{"id":1930,"date":"2019-04-15T12:22:49","date_gmt":"2019-04-15T12:22:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&#038;p=1930"},"modified":"2020-03-02T01:02:25","modified_gmt":"2020-03-02T01:02:25","slug":"4-4-root-functions-and-their-transformation","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/4-4-root-functions-and-their-transformation\/","title":{"raw":"4.4 Root Functions and Their Transformations","rendered":"4.4 Root Functions and Their Transformations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Identify characteristic of odd and even root functions.<\/li>\r\n \t<li>Determine the properties of transformed root functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\nA\u00a0root function\u00a0is a power function of the form [latex]f\\left(x\\right)=x^\\frac{1}{n}[\/latex], where [latex]n[\/latex] is a positive integer greater than one.\u00a0 For example,\u00a0[latex]f\\left(x\\right)=x^\\frac{1}{2}=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex] is the square-root function and\u00a0\u00a0[latex]g\\left(x\\right)=x^\\frac{1}{3}=\\sqrt[\\leftroot{1}\\uproot{2}3]{x}[\/latex] is the cube-root functions.\r\n\r\nThe root functions\u00a0[latex]f\\left(x\\right)=x^\\frac{1}{n}[\/latex] have defining characteristics depending on whether [latex]n[\/latex] is odd or even.\u00a0 For all positive even integers [latex]n\\geq2[\/latex], the domain of\u00a0 [latex]f\\left(x\\right)=x^\\frac{1}{n}[\/latex] is the interval [latex]\\left[0,\\infty\\right).[\/latex]\u00a0\u00a0Figure 1 shows the the functions [latex]f\\left(x\\right)=x^\\frac{1}{2}=\\sqrt[\\leftroot{1}\\uproot{2} ]{x},[\/latex]\u00a0 [latex]g\\left(x\\right)=x^\\frac{1}{4}=\\sqrt[\\leftroot{1}\\uproot{2}4]{x}[\/latex] and [latex]h\\left(x\\right)=x^\\frac{1}{6}=\\sqrt[\\leftroot{1}\\uproot{2}6]{x}[\/latex] which are all even root functions.\u00a0\u00a0<a id=\"Figure 4_4_1\"><\/a>\r\n\r\n[caption id=\"attachment_1949\" align=\"aligncenter\" width=\"791\"]<img class=\"wp-image-1949 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/15194834\/Even-Root-Function-Graphic.png\" alt=\"Graphic comparing even root functions\" width=\"791\" height=\"395\" \/> Figure 1[\/caption]\r\n\r\nNotice that these graphs have similar shapes, very much like that of the square root function in the toolkit. However, as the value of n increases, the graphs steepen somewhat near the origin and become flatter away from the origin growing more slowly.\u00a0 The [latex]x[\/latex] and [latex]y[\/latex] intercepts of these functions are [latex]\\left(0,0\\right)[\/latex]. The end behavior for the even root function only makes sense as [latex]x[\/latex] increases without bound since negative values are not in the domain.\u00a0 We observe as [latex]x\\to\\infty,\\textrm{ }f\\left(x\\right)\\to\\infty[\/latex].\r\n\r\nFor all positive odd integers [latex]n\\geq3[\/latex], the domain of\u00a0 [latex]f\\left(x\\right)=x^\\frac{1}{n}[\/latex] is the set of all real numbers.\u00a0 Since [latex]x^\\frac{1}{n}=\\left(-x\\right)^\\frac{1}{n}[\/latex] for positive odd integers [latex]n[\/latex],\u00a0[latex]f\\left(x\\right)=x^\\frac{1}{n}[\/latex] is an odd function if [latex]n[\/latex] is a positive odd number. Figure 2 shows the functions [latex]f\\left(x\\right)=x^\\frac{1}{3}=\\sqrt[\\leftroot{1}\\uproot{2}3]{x},[\/latex] [latex]g\\left(x\\right)=x^\\frac{1}{5}=\\sqrt[\\leftroot{1}\\uproot{2}5]{x}[\/latex] and [latex]h\\left(x\\right)=x^\\frac{1}{7}=\\sqrt[\\leftroot{1}\\uproot{2}7]{x}[\/latex] which are all odd root functions.<a id=\"Figure 4_4_2\"><\/a>\r\n\r\n[caption id=\"attachment_1948\" align=\"aligncenter\" width=\"779\"]<img class=\"wp-image-1948 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/15193632\/Odd-Root-Function-Graphic.png\" alt=\"Graphic compares odd root functions.\" width=\"779\" height=\"371\" \/> Figure 2[\/caption]\r\n\r\nNotice that these graphs look similar to the cube root function in the toolkit. Again, as the value of n increases, the graphs steepens near the origin and become flatter away from the origin increasing more slowly.\u00a0\u00a0The [latex]x[\/latex] and [latex]y[\/latex] intercepts of these functions are [latex]\\left(0,0\\right)[\/latex].\u00a0 The end behavior for the even root function is expressed as\u00a0[latex]x\\to\\infty,\\textrm{ }f\\left(x\\right)\\to\\infty[\/latex] for large values of [latex]x[\/latex] and as\u00a0[latex]x\\to-\\infty,\\textrm{ }f\\left(x\\right)\\to-\\infty[\/latex] for very negative values of [latex]x.[\/latex]\r\n<h3>Transformations of Root Functions<\/h3>\r\nFor transformations of even root functions, the domain and range are effected by horizontal and vertical shifts, reflections and stretches.\u00a0 There are two methods you can use to find the domain.\u00a0 The first method is to use algebra and the idea that even root functions must have non-negative values under the root symbol.\u00a0 The expression under the root symbol is set greater than or equal to zero and the inequality is solved to find the domain.\u00a0 Alternatively, you can use the properties of the transformation by identifying the basic function and determining where the point (0,0) gets transformed to in the new function.\u00a0 The x-coordinate will be the starting or ending point for the domain.\u00a0 If there is not a horizontal reflection, the domain will be from that value to the right and if there is a horizontal reflection, then the domain will go from that value to the left.\r\n\r\nThe range is determined by identifying the basic function and determining what transformation is applied to get the function you are working with.\u00a0 After applying transformation to the point (0,0), the y-coordinate tells you where the range starts or ends.\u00a0 If there is not a vertical reflection the range will be from that value to infinity and if there is a vertical reflection the range will be from minus infinity to that value.\r\n<div class=\"textbox examples\">\r\n<h3>How To<\/h3>\r\n<strong>Given a root function, find the domain and range.<\/strong>\r\n\r\n<em>Domain Method 1: Algebraically<\/em>\r\n<ol>\r\n \t<li>Set the expression under the root symbol greater than or equal to zero and solve.<\/li>\r\n \t<li>Write the solution in interval notation.\u00a0 Remember to use the square bracket as appropriate.<\/li>\r\n<\/ol>\r\n<em>Domain Method 2: Transformations<\/em>\r\n<ol>\r\n \t<li>Identify the basic root function.<\/li>\r\n \t<li>Describe the transformation in words and then determine where the point (0,0) gets mapped to under that transformation.<\/li>\r\n \t<li>If there is not a vertical reflection, the domain is from the x-coordinate of the transformed point to infinity.\u00a0 If there is a vertical reflection, the domain is from minus infinity to that x-coordinate.<\/li>\r\n<\/ol>\r\n<em>Range<\/em>\r\n<ol>\r\n \t<li>Identify the basic root function.<\/li>\r\n \t<li>Describe the transformation in words and then determine where the point (0,0) gets mapped to under that transformation.<\/li>\r\n \t<li>The y-coordinate tells you where the range starts or ends.\u00a0 If there is not a vertical reflection the range will be from that value to infinity and if there is a vertical reflection the range will be from minus infinity to that value.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1:\u00a0 The Domain and Range of an Even Root Function<\/h3>\r\nFind the domain, range and intercepts of the square root function shifted 3 units left and 1 unit up.\r\n\r\n[reveal-answer q=\"35558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"35558\"]\r\n\r\nFirst, find the equation for the function.\u00a0 [latex]f\\left(x+3\\right)+1=\\sqrt[\\leftroot{1}\\uproot{2} ]{x+3}+1.[\/latex]\r\n<p style=\"text-align: left\"><em>Method 1:<\/em>\u00a0 The domain of an even root function must have non-negative values under the root symbol, so we solve the inequality<\/p>\r\n<p style=\"text-align: center\">[latex]x+3\\geq0.[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]x\\geq-3[\/latex]<\/p>\r\nTherefore, the domain is all real numbers greater that or equal to negative 3 or in interval notation [latex]\\left[-3,\\infty\\right).[\/latex]\r\n\r\n<em>Method 2:<\/em>\u00a0\u00a0Alternatively, the point [latex]\\left(0,0\\right)[\/latex] is shifted to the point\u00a0[latex]\\left(-3,1\\right).[\/latex]\u00a0 The starting value for the domain is -3 and since there is no horizontal reflection the graph opens to the right like [latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{x}.[\/latex]\u00a0 Again the domain is\u00a0[latex]\\left[-3,\\infty\\right).[\/latex]\r\n\r\n&nbsp;\r\n\r\nThe range of the square root\u00a0will be shifted up one unit, so the range is all real numbers greater than or equal to one or in interval notation [latex]\\left[1,\\infty\\right).[\/latex]\u00a0 Notice that the starting value of 1 is reflected in the shift of the point [latex]\\left(0,0\\right)[\/latex] above and since there is no vertical reflection the interval goes in the direction of infinity.\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2:\u00a0 Domain and Intercepts of Even Root Functions<\/h3>\r\nFind the domain and range for\r\n<p style=\"padding-left: 30px\">a. [latex]g\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}4]{3-2x}.[\/latex]<\/p>\r\n<p style=\"padding-left: 30px\">b.\u00a0[latex]h\\left(x\\right)=-3\\sqrt[\\leftroot{1}\\uproot{2}4]{x}.[\/latex]<\/p>\r\n[reveal-answer q=\"387766\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"387766\"]\r\n\r\na. <em>Method 1:<\/em> Even root functions have non-negative input, so the domain of\u00a0[latex]g\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}4]{3-2x}[\/latex] is found by solving the inequality\r\n<p style=\"text-align: center\">[latex]3-2x\\geq0.[\/latex]<\/p>\r\nThe solution is\r\n<p style=\"text-align: center\">[latex]x\\leq\\frac{3}{2}[\/latex].<\/p>\r\nTherefore the domain is [latex]\\left(-\\infty,1.5\\right].[\/latex]\r\n\r\n<em>Method 2:<\/em>\u00a0 The function is a horizontal shift of [latex]\\sqrt[\\leftroot{1}\\uproot{2}4]{x}[\/latex] left by 3 units followed by a horizontal compression by a factor of 1\/2 and then a horizontal reflection.\u00a0 The point (0,0) is mapped as follows:\r\n<p style=\"text-align: center\">[latex]\\left(0,0\\right)\\to\\left(-3, 0\\right)\\to\\left(-1.5,0\\right)\\to\\left(1.5,0\\right)[\/latex]<\/p>\r\nIt is the horizontal reflection that makes the graph open to the left rather than the right.\r\n\r\nThe domain is [latex]\\left(-\\infty,1.5\\right].[\/latex]\r\n\r\nThe range remains\u00a0[latex]\\left(0,\\infty\\right)[\/latex] since there are no vertical transformations.\r\n\r\nThe domain and range can be seen in the graph below.\r\n\r\n<img class=\"aligncenter wp-image-1981 size-full\" style=\"font-size: 16px\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/19010943\/44Ex1.png\" alt=\"\" width=\"998\" height=\"464\" \/>\r\nb.\u00a0\u00a0Notice that [latex]h\\left(x\\right)=-3\\sqrt[\\leftroot{1}\\uproot{2}4]{x}[\/latex] is\u00a0a vertical reflection and stretch by a factor of 3 of the function [latex]\\sqrt[\\leftroot{1}\\uproot{2}4]{x}.[\/latex]\u00a0 This tells us that the properties associated with the output value will change so we need to consider the range carefully.\r\n\r\nThe domain will be [latex]\\left [0,\\infty\\right) [\/latex] since there are no horizontal transformations.\r\n\r\nThe vertical reflection does effect the range.\u00a0 Since the fourth root function outputs positive numbers or zero, when we multiply by a negative number will will have negative values or zero.\u00a0 Therefore, the range is [latex]\\left(-\\infty,0\\right][\/latex].\r\n\r\nThe vertical reflection and range is shown in the graph below.\r\n\r\n<img class=\"aligncenter wp-image-1982 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/19011252\/44Ex2-1024x448.png\" alt=\"\" width=\"1024\" height=\"448\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div><\/div>\r\n<h3>Intercepts of Even Root Functions<\/h3>\r\nTransformations of even root functions may or may not have [latex]x[\/latex] or [latex]y[\/latex] intercepts.\u00a0 If [latex]x = 0[\/latex] is in the domain of the transformed function then there will be a y-intercept found by evaluating [latex]f\\left(0\\right).[\/latex]\u00a0 If [latex]y=0[\/latex] is in the range, then there will be an x-intercept and we solve [latex]f\\left(x\\right)=0.[\/latex]\r\n<div class=\"textbox examples\">\r\n<h3>Example 3:\u00a0 Intercepts of Transformations of Even Root Functions<\/h3>\r\nFind x-intercepts and y-intercepts for\r\n<p style=\"padding-left: 30px\">a.\u00a0[latex]g\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}4]{3-2x}.[\/latex]<\/p>\r\n<p style=\"padding-left: 30px\">b. [latex]h\\left(x\\right)=-3\\sqrt[\\leftroot{1}\\uproot{2}4]{x}.[\/latex]<\/p>\r\n<p style=\"padding-left: 30px\">c.\u00a0[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x+3}+1.[\/latex]<\/p>\r\n[reveal-answer q=\"143651\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"143651\"]\r\n\r\na.\u00a0\u00a0The x-intercept has [latex]y=0[\/latex].\u00a0 We solve the equation [latex]\\sqrt[\\leftroot{1}\\uproot{2}4]{3-2x}=0[\/latex] by raising both sides to the 4<sup>th<\/sup> power to get [latex]3-2x=0.[\/latex] Finally, [latex]x=1.5.[\/latex]\u00a0 The x-intercept is [latex]\\left(1.5,0\\right).[\/latex]\r\n\r\nFor the y-intercept, [latex]f\\left(0\\right)[\/latex] is evaluated.\u00a0 We get\u00a0[latex]f\\left(0\\right)=\\sqrt[\\leftroot{1}\\uproot{2}4]{3-2\\left(0\\right)}=\\sqrt[\\leftroot{1}\\uproot{2}4]{3}\\approx1.316.[\/latex] Therefore, the y-intercept is [latex]\\left(0, 1.316\\right).[\/latex]\r\n\r\n&nbsp;\r\n\r\nb. Since there is no horizontal or vertical shift both the [latex]x[\/latex] and [latex]y[\/latex] intercepts are [latex]\\left(0,0\\right).[\/latex]\r\n\r\n&nbsp;\r\n\r\nc.\u00a0\u00a0Evaluate [latex]f\\left(0\\right)[\/latex] to get [latex]f\\left(0\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{0+3}+1\\approx2.73[\/latex].\u00a0 The y-intercept is approximately [latex]\\left(0,2.73\\right).[\/latex]\r\n\r\nSince [latex]y=0[\/latex] is not in the range because the graph was shifted up one unit, there is no x-intercept.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It #1<\/h3>\r\nFind the domain, range and intercepts of the fourth root function shifted 2 units right and 1 unit down.\r\n\r\n[reveal-answer q=\"190017\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"190017\"]\r\n\r\nThe domain is\u00a0[latex]\\left[2,\\infty\\right).[\/latex]\u00a0 The range is\u00a0[latex]\\left[-1,\\infty\\right).[\/latex]\u00a0 There is no y-intercept because [latex]x=0[\/latex] is not in the domain.\u00a0 The x-intercept is\u00a0[latex]\\left[3,0\\right).[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It #2<\/h3>\r\nFind the domain, x-intercepts and y-intercepts for\u00a0[latex]g\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{7-0.5x}.[\/latex]\r\n\r\n[reveal-answer q=\"161370\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161370\"]\r\n\r\nThe domain is\u00a0[latex]\\left(-\\infty,14\\right].[\/latex] The x-intercept is [latex]\\left(14,0\\right)[\/latex] and the y-intercept is [latex]\\left(0,2.646\\right).[\/latex] These values can be confirmed in the graph below.\r\n<p style=\"text-align: center\"><img class=\"aligncenter wp-image-1962 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/17201742\/44Try2.png\" alt=\"\" width=\"1010\" height=\"469\" \/><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>End Behavior of Even Root\u00a0 Functions<\/h3>\r\nThe final property to examine for even root functions and their transformations is the end or long term behavior.\u00a0 Since the domain is only part of the real numbers only behavior to the left or right needs to be determined depending on whether the domain goes toward minus infinity or plus infinity.\r\n<div class=\"textbox examples\">\r\n<h3>Example 4:\u00a0 End Behavior of a Horizontally Reflected Even Root Function<\/h3>\r\nDetermine the end behavior of\u00a0[latex]k\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}6]{2-x}.[\/latex]\r\n\r\n[reveal-answer q=\"557948\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"557948\"]\r\n\r\n[latex]k\\left(x\\right)[\/latex] is a shift left by 2 units of\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}6]{x}[\/latex] followed by a horizontal reflection.\u00a0 The shift by 2 units will not effect the end behavior but the reflection will make it so the graph opens to the left.\u00a0 Since (0,0) transforms as follows\r\n<p style=\"text-align: center\">[latex]\\left(0,0\\right)\\to\\left(-2,0\\right)\\to\\left(2,0\\right)[\/latex]<\/p>\r\nthe domain is [latex]\\left(-\\infty,2\\right)[\/latex].\u00a0 Right end behavior does not make sense for this function.\r\n\r\nAs [latex]x[\/latex] goes further and further to the left the output will become larger and larger.\u00a0 We write this as [latex]x\\to-\\infty, f\\left(x\\right)\\to\\infty.[\/latex]\r\n\r\nWe can confirm this in the table and graph below.\u00a0 For the table, even when we chose extremely negative values for x, the output is not really large but we see that it continues to get bigger and does not level off.\r\n<table class=\"lines\" style=\"border-collapse: collapse;width: 32.0386%;height: 133px\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 50%;text-align: center\">[latex]x[\/latex]<\/td>\r\n<td style=\"width: 50%;text-align: center\">[latex]f\\left(x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;text-align: center\">- 100<\/td>\r\n<td style=\"width: 50%;text-align: center\">2.16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;text-align: center\">- 1,000<\/td>\r\n<td style=\"width: 50%;text-align: center\">3.16<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;text-align: center\">- 10,000<\/td>\r\n<td style=\"width: 50%;text-align: center\">4.64<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;text-align: center\">- 1,000,000<\/td>\r\n<td style=\"width: 50%;text-align: center\">10<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;text-align: center\">- 1,000,000,000<\/td>\r\n<td style=\"width: 50%;text-align: center\">31.6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<img class=\"aligncenter wp-image-1983 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/19011412\/44Ex3.png\" alt=\"\" width=\"967\" height=\"476\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Properties of Odd Root Functions<\/h3>\r\nOdd root functions do not have their domains and ranges change under transformations since they are defined on [latex]\\left(-\\infty,\\infty\\right).[\/latex]\u00a0 However with horizontal and vertical shifts, the intercepts are expected to change and if there is a horizontal or vertical reflection, the end behavior may be effected.\r\n<div class=\"textbox examples\">\r\n<h3>Example 5:\u00a0 Properties of a Reflected Odd Root Function<\/h3>\r\nDetermine the domain, range, x-intercept, y-intercept and end behavior of the function\u00a0[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{3-x}+1.[\/latex]\r\n\r\n[reveal-answer q=\"505068\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"505068\"]\r\n\r\nThis equation is a shift of\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}3]{x}[\/latex] left by 3 units and then a horizontal reflection.\u00a0 Finally the graph is shifted up 1 unit.\r\n\r\n&nbsp;\r\n\r\nSince the domain and range of\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}3]{x}[\/latex] is all real numbers, these transformations will not effect these properties.\u00a0 Therefore the domain and range of [latex]k\\left(x\\right)[\/latex] are [latex]\\left(-\\infty,\\infty\\right).[\/latex]\r\n\r\n&nbsp;\r\n\r\nTo find the x-intercept we solve the equation\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}3]{3-x}+1=0.[\/latex] Begin by subtracting one from each side to get\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}3]{3-x}=-1.[\/latex]\u00a0 Next, cube both sides to get\u00a0[latex]3-x=-1.[\/latex]\u00a0 Finally [latex]x=4.[\/latex]\u00a0 The x-intercept is [latex]\\left(4,0\\right).[\/latex]\r\n\r\n&nbsp;\r\n\r\nTo find the y-intercept we evaluate [latex]f\\left(0\\right).[\/latex]\u00a0 We get\u00a0[latex]f\\left(0\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{3-0}+1=\\sqrt[\\leftroot{1}\\uproot{2}3]{3}+1\\approx2.442.[\/latex] The y-intercept is [latex]\\left(0,2.442\\right).[\/latex]\r\n\r\n&nbsp;\r\n\r\nThe end behavior will be similar to\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}3]{-x}\\textrm{ or }-\\sqrt[\\leftroot{1}\\uproot{2}3]{x}[\/latex] since odd root functions are odd.\u00a0 Therefore, as [latex]x[\/latex] becomes more and more negative, [latex]f\\left(x\\right)[\/latex] increases without bound and as [latex]x[\/latex] becomes larger and larger [latex]f\\left(x\\right)[\/latex] decreases without bound.\u00a0 We write, as [latex]x\\to-\\infty, f\\left(x\\right)\\to\\infty[\/latex] and [latex]x\\to\\infty, f\\left(x\\right)\\to-\\infty.[\/latex]\r\n\r\n&nbsp;\r\n\r\nThese properties can be confirmed in the graph below.\r\n\r\n<img class=\"aligncenter wp-image-1965 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/17210437\/44Ex5.png\" alt=\"\" width=\"990\" height=\"457\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul>\r\n \t<li>Root functions are steep near the origin and then grow slowly.<\/li>\r\n \t<li>The domain, range, intercepts and end behavior may change when even root functions are transformed.\r\n<ul>\r\n \t<li>Intercepts may not exist for all transformed even root functions.<\/li>\r\n \t<li>Only one side of end behavior makes sense for transformed even root functions.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The domain and range for transformed odd root functions remains [latex]\\left(-\\infty,\\infty\\right)[\/latex]<\/li>\r\n \t<li>The intercepts and end behavior may change when odd root functions are transformed.\u00a0 However, there will and [latex]x[\/latex] and [latex]y[\/latex] intercept and the end behavior must be considered on both the right and left.<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Identify characteristic of odd and even root functions.<\/li>\n<li>Determine the properties of transformed root functions.<\/li>\n<\/ul>\n<\/div>\n<p>A\u00a0root function\u00a0is a power function of the form [latex]f\\left(x\\right)=x^\\frac{1}{n}[\/latex], where [latex]n[\/latex] is a positive integer greater than one.\u00a0 For example,\u00a0[latex]f\\left(x\\right)=x^\\frac{1}{2}=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex] is the square-root function and\u00a0\u00a0[latex]g\\left(x\\right)=x^\\frac{1}{3}=\\sqrt[\\leftroot{1}\\uproot{2}3]{x}[\/latex] is the cube-root functions.<\/p>\n<p>The root functions\u00a0[latex]f\\left(x\\right)=x^\\frac{1}{n}[\/latex] have defining characteristics depending on whether [latex]n[\/latex] is odd or even.\u00a0 For all positive even integers [latex]n\\geq2[\/latex], the domain of\u00a0 [latex]f\\left(x\\right)=x^\\frac{1}{n}[\/latex] is the interval [latex]\\left[0,\\infty\\right).[\/latex]\u00a0\u00a0Figure 1 shows the the functions [latex]f\\left(x\\right)=x^\\frac{1}{2}=\\sqrt[\\leftroot{1}\\uproot{2} ]{x},[\/latex]\u00a0 [latex]g\\left(x\\right)=x^\\frac{1}{4}=\\sqrt[\\leftroot{1}\\uproot{2}4]{x}[\/latex] and [latex]h\\left(x\\right)=x^\\frac{1}{6}=\\sqrt[\\leftroot{1}\\uproot{2}6]{x}[\/latex] which are all even root functions.\u00a0\u00a0<a id=\"Figure 4_4_1\"><\/a><\/p>\n<div id=\"attachment_1949\" style=\"width: 801px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1949\" class=\"wp-image-1949 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/15194834\/Even-Root-Function-Graphic.png\" alt=\"Graphic comparing even root functions\" width=\"791\" height=\"395\" \/><\/p>\n<p id=\"caption-attachment-1949\" class=\"wp-caption-text\">Figure 1<\/p>\n<\/div>\n<p>Notice that these graphs have similar shapes, very much like that of the square root function in the toolkit. However, as the value of n increases, the graphs steepen somewhat near the origin and become flatter away from the origin growing more slowly.\u00a0 The [latex]x[\/latex] and [latex]y[\/latex] intercepts of these functions are [latex]\\left(0,0\\right)[\/latex]. The end behavior for the even root function only makes sense as [latex]x[\/latex] increases without bound since negative values are not in the domain.\u00a0 We observe as [latex]x\\to\\infty,\\textrm{ }f\\left(x\\right)\\to\\infty[\/latex].<\/p>\n<p>For all positive odd integers [latex]n\\geq3[\/latex], the domain of\u00a0 [latex]f\\left(x\\right)=x^\\frac{1}{n}[\/latex] is the set of all real numbers.\u00a0 Since [latex]x^\\frac{1}{n}=\\left(-x\\right)^\\frac{1}{n}[\/latex] for positive odd integers [latex]n[\/latex],\u00a0[latex]f\\left(x\\right)=x^\\frac{1}{n}[\/latex] is an odd function if [latex]n[\/latex] is a positive odd number. Figure 2 shows the functions [latex]f\\left(x\\right)=x^\\frac{1}{3}=\\sqrt[\\leftroot{1}\\uproot{2}3]{x},[\/latex] [latex]g\\left(x\\right)=x^\\frac{1}{5}=\\sqrt[\\leftroot{1}\\uproot{2}5]{x}[\/latex] and [latex]h\\left(x\\right)=x^\\frac{1}{7}=\\sqrt[\\leftroot{1}\\uproot{2}7]{x}[\/latex] which are all odd root functions.<a id=\"Figure 4_4_2\"><\/a><\/p>\n<div id=\"attachment_1948\" style=\"width: 789px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1948\" class=\"wp-image-1948 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/15193632\/Odd-Root-Function-Graphic.png\" alt=\"Graphic compares odd root functions.\" width=\"779\" height=\"371\" \/><\/p>\n<p id=\"caption-attachment-1948\" class=\"wp-caption-text\">Figure 2<\/p>\n<\/div>\n<p>Notice that these graphs look similar to the cube root function in the toolkit. Again, as the value of n increases, the graphs steepens near the origin and become flatter away from the origin increasing more slowly.\u00a0\u00a0The [latex]x[\/latex] and [latex]y[\/latex] intercepts of these functions are [latex]\\left(0,0\\right)[\/latex].\u00a0 The end behavior for the even root function is expressed as\u00a0[latex]x\\to\\infty,\\textrm{ }f\\left(x\\right)\\to\\infty[\/latex] for large values of [latex]x[\/latex] and as\u00a0[latex]x\\to-\\infty,\\textrm{ }f\\left(x\\right)\\to-\\infty[\/latex] for very negative values of [latex]x.[\/latex]<\/p>\n<h3>Transformations of Root Functions<\/h3>\n<p>For transformations of even root functions, the domain and range are effected by horizontal and vertical shifts, reflections and stretches.\u00a0 There are two methods you can use to find the domain.\u00a0 The first method is to use algebra and the idea that even root functions must have non-negative values under the root symbol.\u00a0 The expression under the root symbol is set greater than or equal to zero and the inequality is solved to find the domain.\u00a0 Alternatively, you can use the properties of the transformation by identifying the basic function and determining where the point (0,0) gets transformed to in the new function.\u00a0 The x-coordinate will be the starting or ending point for the domain.\u00a0 If there is not a horizontal reflection, the domain will be from that value to the right and if there is a horizontal reflection, then the domain will go from that value to the left.<\/p>\n<p>The range is determined by identifying the basic function and determining what transformation is applied to get the function you are working with.\u00a0 After applying transformation to the point (0,0), the y-coordinate tells you where the range starts or ends.\u00a0 If there is not a vertical reflection the range will be from that value to infinity and if there is a vertical reflection the range will be from minus infinity to that value.<\/p>\n<div class=\"textbox examples\">\n<h3>How To<\/h3>\n<p><strong>Given a root function, find the domain and range.<\/strong><\/p>\n<p><em>Domain Method 1: Algebraically<\/em><\/p>\n<ol>\n<li>Set the expression under the root symbol greater than or equal to zero and solve.<\/li>\n<li>Write the solution in interval notation.\u00a0 Remember to use the square bracket as appropriate.<\/li>\n<\/ol>\n<p><em>Domain Method 2: Transformations<\/em><\/p>\n<ol>\n<li>Identify the basic root function.<\/li>\n<li>Describe the transformation in words and then determine where the point (0,0) gets mapped to under that transformation.<\/li>\n<li>If there is not a vertical reflection, the domain is from the x-coordinate of the transformed point to infinity.\u00a0 If there is a vertical reflection, the domain is from minus infinity to that x-coordinate.<\/li>\n<\/ol>\n<p><em>Range<\/em><\/p>\n<ol>\n<li>Identify the basic root function.<\/li>\n<li>Describe the transformation in words and then determine where the point (0,0) gets mapped to under that transformation.<\/li>\n<li>The y-coordinate tells you where the range starts or ends.\u00a0 If there is not a vertical reflection the range will be from that value to infinity and if there is a vertical reflection the range will be from minus infinity to that value.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 1:\u00a0 The Domain and Range of an Even Root Function<\/h3>\n<p>Find the domain, range and intercepts of the square root function shifted 3 units left and 1 unit up.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q35558\">Show Solution<\/span><\/p>\n<div id=\"q35558\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, find the equation for the function.\u00a0 [latex]f\\left(x+3\\right)+1=\\sqrt[\\leftroot{1}\\uproot{2} ]{x+3}+1.[\/latex]<\/p>\n<p style=\"text-align: left\"><em>Method 1:<\/em>\u00a0 The domain of an even root function must have non-negative values under the root symbol, so we solve the inequality<\/p>\n<p style=\"text-align: center\">[latex]x+3\\geq0.[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]x\\geq-3[\/latex]<\/p>\n<p>Therefore, the domain is all real numbers greater that or equal to negative 3 or in interval notation [latex]\\left[-3,\\infty\\right).[\/latex]<\/p>\n<p><em>Method 2:<\/em>\u00a0\u00a0Alternatively, the point [latex]\\left(0,0\\right)[\/latex] is shifted to the point\u00a0[latex]\\left(-3,1\\right).[\/latex]\u00a0 The starting value for the domain is -3 and since there is no horizontal reflection the graph opens to the right like [latex]\\sqrt[\\leftroot{1}\\uproot{2} ]{x}.[\/latex]\u00a0 Again the domain is\u00a0[latex]\\left[-3,\\infty\\right).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The range of the square root\u00a0will be shifted up one unit, so the range is all real numbers greater than or equal to one or in interval notation [latex]\\left[1,\\infty\\right).[\/latex]\u00a0 Notice that the starting value of 1 is reflected in the shift of the point [latex]\\left(0,0\\right)[\/latex] above and since there is no vertical reflection the interval goes in the direction of infinity.<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2:\u00a0 Domain and Intercepts of Even Root Functions<\/h3>\n<p>Find the domain and range for<\/p>\n<p style=\"padding-left: 30px\">a. [latex]g\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}4]{3-2x}.[\/latex]<\/p>\n<p style=\"padding-left: 30px\">b.\u00a0[latex]h\\left(x\\right)=-3\\sqrt[\\leftroot{1}\\uproot{2}4]{x}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q387766\">Show Solution<\/span><\/p>\n<div id=\"q387766\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. <em>Method 1:<\/em> Even root functions have non-negative input, so the domain of\u00a0[latex]g\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}4]{3-2x}[\/latex] is found by solving the inequality<\/p>\n<p style=\"text-align: center\">[latex]3-2x\\geq0.[\/latex]<\/p>\n<p>The solution is<\/p>\n<p style=\"text-align: center\">[latex]x\\leq\\frac{3}{2}[\/latex].<\/p>\n<p>Therefore the domain is [latex]\\left(-\\infty,1.5\\right].[\/latex]<\/p>\n<p><em>Method 2:<\/em>\u00a0 The function is a horizontal shift of [latex]\\sqrt[\\leftroot{1}\\uproot{2}4]{x}[\/latex] left by 3 units followed by a horizontal compression by a factor of 1\/2 and then a horizontal reflection.\u00a0 The point (0,0) is mapped as follows:<\/p>\n<p style=\"text-align: center\">[latex]\\left(0,0\\right)\\to\\left(-3, 0\\right)\\to\\left(-1.5,0\\right)\\to\\left(1.5,0\\right)[\/latex]<\/p>\n<p>It is the horizontal reflection that makes the graph open to the left rather than the right.<\/p>\n<p>The domain is [latex]\\left(-\\infty,1.5\\right].[\/latex]<\/p>\n<p>The range remains\u00a0[latex]\\left(0,\\infty\\right)[\/latex] since there are no vertical transformations.<\/p>\n<p>The domain and range can be seen in the graph below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1981 size-full\" style=\"font-size: 16px\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/19010943\/44Ex1.png\" alt=\"\" width=\"998\" height=\"464\" \/><br \/>\nb.\u00a0\u00a0Notice that [latex]h\\left(x\\right)=-3\\sqrt[\\leftroot{1}\\uproot{2}4]{x}[\/latex] is\u00a0a vertical reflection and stretch by a factor of 3 of the function [latex]\\sqrt[\\leftroot{1}\\uproot{2}4]{x}.[\/latex]\u00a0 This tells us that the properties associated with the output value will change so we need to consider the range carefully.<\/p>\n<p>The domain will be [latex]\\left [0,\\infty\\right)[\/latex] since there are no horizontal transformations.<\/p>\n<p>The vertical reflection does effect the range.\u00a0 Since the fourth root function outputs positive numbers or zero, when we multiply by a negative number will will have negative values or zero.\u00a0 Therefore, the range is [latex]\\left(-\\infty,0\\right][\/latex].<\/p>\n<p>The vertical reflection and range is shown in the graph below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1982 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/19011252\/44Ex2-1024x448.png\" alt=\"\" width=\"1024\" height=\"448\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<h3>Intercepts of Even Root Functions<\/h3>\n<p>Transformations of even root functions may or may not have [latex]x[\/latex] or [latex]y[\/latex] intercepts.\u00a0 If [latex]x = 0[\/latex] is in the domain of the transformed function then there will be a y-intercept found by evaluating [latex]f\\left(0\\right).[\/latex]\u00a0 If [latex]y=0[\/latex] is in the range, then there will be an x-intercept and we solve [latex]f\\left(x\\right)=0.[\/latex]<\/p>\n<div class=\"textbox examples\">\n<h3>Example 3:\u00a0 Intercepts of Transformations of Even Root Functions<\/h3>\n<p>Find x-intercepts and y-intercepts for<\/p>\n<p style=\"padding-left: 30px\">a.\u00a0[latex]g\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}4]{3-2x}.[\/latex]<\/p>\n<p style=\"padding-left: 30px\">b. [latex]h\\left(x\\right)=-3\\sqrt[\\leftroot{1}\\uproot{2}4]{x}.[\/latex]<\/p>\n<p style=\"padding-left: 30px\">c.\u00a0[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x+3}+1.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q143651\">Show Solution<\/span><\/p>\n<div id=\"q143651\" class=\"hidden-answer\" style=\"display: none\">\n<p>a.\u00a0\u00a0The x-intercept has [latex]y=0[\/latex].\u00a0 We solve the equation [latex]\\sqrt[\\leftroot{1}\\uproot{2}4]{3-2x}=0[\/latex] by raising both sides to the 4<sup>th<\/sup> power to get [latex]3-2x=0.[\/latex] Finally, [latex]x=1.5.[\/latex]\u00a0 The x-intercept is [latex]\\left(1.5,0\\right).[\/latex]<\/p>\n<p>For the y-intercept, [latex]f\\left(0\\right)[\/latex] is evaluated.\u00a0 We get\u00a0[latex]f\\left(0\\right)=\\sqrt[\\leftroot{1}\\uproot{2}4]{3-2\\left(0\\right)}=\\sqrt[\\leftroot{1}\\uproot{2}4]{3}\\approx1.316.[\/latex] Therefore, the y-intercept is [latex]\\left(0, 1.316\\right).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>b. Since there is no horizontal or vertical shift both the [latex]x[\/latex] and [latex]y[\/latex] intercepts are [latex]\\left(0,0\\right).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>c.\u00a0\u00a0Evaluate [latex]f\\left(0\\right)[\/latex] to get [latex]f\\left(0\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{0+3}+1\\approx2.73[\/latex].\u00a0 The y-intercept is approximately [latex]\\left(0,2.73\\right).[\/latex]<\/p>\n<p>Since [latex]y=0[\/latex] is not in the range because the graph was shifted up one unit, there is no x-intercept.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It #1<\/h3>\n<p>Find the domain, range and intercepts of the fourth root function shifted 2 units right and 1 unit down.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q190017\">Show Solution<\/span><\/p>\n<div id=\"q190017\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is\u00a0[latex]\\left[2,\\infty\\right).[\/latex]\u00a0 The range is\u00a0[latex]\\left[-1,\\infty\\right).[\/latex]\u00a0 There is no y-intercept because [latex]x=0[\/latex] is not in the domain.\u00a0 The x-intercept is\u00a0[latex]\\left[3,0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It #2<\/h3>\n<p>Find the domain, x-intercepts and y-intercepts for\u00a0[latex]g\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{7-0.5x}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q161370\">Show Solution<\/span><\/p>\n<div id=\"q161370\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is\u00a0[latex]\\left(-\\infty,14\\right].[\/latex] The x-intercept is [latex]\\left(14,0\\right)[\/latex] and the y-intercept is [latex]\\left(0,2.646\\right).[\/latex] These values can be confirmed in the graph below.<\/p>\n<p style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1962 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/17201742\/44Try2.png\" alt=\"\" width=\"1010\" height=\"469\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>End Behavior of Even Root\u00a0 Functions<\/h3>\n<p>The final property to examine for even root functions and their transformations is the end or long term behavior.\u00a0 Since the domain is only part of the real numbers only behavior to the left or right needs to be determined depending on whether the domain goes toward minus infinity or plus infinity.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 4:\u00a0 End Behavior of a Horizontally Reflected Even Root Function<\/h3>\n<p>Determine the end behavior of\u00a0[latex]k\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}6]{2-x}.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q557948\">Show Solution<\/span><\/p>\n<div id=\"q557948\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]k\\left(x\\right)[\/latex] is a shift left by 2 units of\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}6]{x}[\/latex] followed by a horizontal reflection.\u00a0 The shift by 2 units will not effect the end behavior but the reflection will make it so the graph opens to the left.\u00a0 Since (0,0) transforms as follows<\/p>\n<p style=\"text-align: center\">[latex]\\left(0,0\\right)\\to\\left(-2,0\\right)\\to\\left(2,0\\right)[\/latex]<\/p>\n<p>the domain is [latex]\\left(-\\infty,2\\right)[\/latex].\u00a0 Right end behavior does not make sense for this function.<\/p>\n<p>As [latex]x[\/latex] goes further and further to the left the output will become larger and larger.\u00a0 We write this as [latex]x\\to-\\infty, f\\left(x\\right)\\to\\infty.[\/latex]<\/p>\n<p>We can confirm this in the table and graph below.\u00a0 For the table, even when we chose extremely negative values for x, the output is not really large but we see that it continues to get bigger and does not level off.<\/p>\n<table class=\"lines\" style=\"border-collapse: collapse;width: 32.0386%;height: 133px\">\n<tbody>\n<tr>\n<td style=\"width: 50%;text-align: center\">[latex]x[\/latex]<\/td>\n<td style=\"width: 50%;text-align: center\">[latex]f\\left(x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;text-align: center\">&#8211; 100<\/td>\n<td style=\"width: 50%;text-align: center\">2.16<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;text-align: center\">&#8211; 1,000<\/td>\n<td style=\"width: 50%;text-align: center\">3.16<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;text-align: center\">&#8211; 10,000<\/td>\n<td style=\"width: 50%;text-align: center\">4.64<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;text-align: center\">&#8211; 1,000,000<\/td>\n<td style=\"width: 50%;text-align: center\">10<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;text-align: center\">&#8211; 1,000,000,000<\/td>\n<td style=\"width: 50%;text-align: center\">31.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1983 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/19011412\/44Ex3.png\" alt=\"\" width=\"967\" height=\"476\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Properties of Odd Root Functions<\/h3>\n<p>Odd root functions do not have their domains and ranges change under transformations since they are defined on [latex]\\left(-\\infty,\\infty\\right).[\/latex]\u00a0 However with horizontal and vertical shifts, the intercepts are expected to change and if there is a horizontal or vertical reflection, the end behavior may be effected.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 5:\u00a0 Properties of a Reflected Odd Root Function<\/h3>\n<p>Determine the domain, range, x-intercept, y-intercept and end behavior of the function\u00a0[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{3-x}+1.[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q505068\">Show Solution<\/span><\/p>\n<div id=\"q505068\" class=\"hidden-answer\" style=\"display: none\">\n<p>This equation is a shift of\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}3]{x}[\/latex] left by 3 units and then a horizontal reflection.\u00a0 Finally the graph is shifted up 1 unit.<\/p>\n<p>&nbsp;<\/p>\n<p>Since the domain and range of\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}3]{x}[\/latex] is all real numbers, these transformations will not effect these properties.\u00a0 Therefore the domain and range of [latex]k\\left(x\\right)[\/latex] are [latex]\\left(-\\infty,\\infty\\right).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>To find the x-intercept we solve the equation\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}3]{3-x}+1=0.[\/latex] Begin by subtracting one from each side to get\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}3]{3-x}=-1.[\/latex]\u00a0 Next, cube both sides to get\u00a0[latex]3-x=-1.[\/latex]\u00a0 Finally [latex]x=4.[\/latex]\u00a0 The x-intercept is [latex]\\left(4,0\\right).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>To find the y-intercept we evaluate [latex]f\\left(0\\right).[\/latex]\u00a0 We get\u00a0[latex]f\\left(0\\right)=\\sqrt[\\leftroot{1}\\uproot{2}3]{3-0}+1=\\sqrt[\\leftroot{1}\\uproot{2}3]{3}+1\\approx2.442.[\/latex] The y-intercept is [latex]\\left(0,2.442\\right).[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The end behavior will be similar to\u00a0[latex]\\sqrt[\\leftroot{1}\\uproot{2}3]{-x}\\textrm{ or }-\\sqrt[\\leftroot{1}\\uproot{2}3]{x}[\/latex] since odd root functions are odd.\u00a0 Therefore, as [latex]x[\/latex] becomes more and more negative, [latex]f\\left(x\\right)[\/latex] increases without bound and as [latex]x[\/latex] becomes larger and larger [latex]f\\left(x\\right)[\/latex] decreases without bound.\u00a0 We write, as [latex]x\\to-\\infty, f\\left(x\\right)\\to\\infty[\/latex] and [latex]x\\to\\infty, f\\left(x\\right)\\to-\\infty.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>These properties can be confirmed in the graph below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1965 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/04\/17210437\/44Ex5.png\" alt=\"\" width=\"990\" height=\"457\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul>\n<li>Root functions are steep near the origin and then grow slowly.<\/li>\n<li>The domain, range, intercepts and end behavior may change when even root functions are transformed.\n<ul>\n<li>Intercepts may not exist for all transformed even root functions.<\/li>\n<li>Only one side of end behavior makes sense for transformed even root functions.<\/li>\n<\/ul>\n<\/li>\n<li>The domain and range for transformed odd root functions remains [latex]\\left(-\\infty,\\infty\\right)[\/latex]<\/li>\n<li>The intercepts and end behavior may change when odd root functions are transformed.\u00a0 However, there will and [latex]x[\/latex] and [latex]y[\/latex] intercept and the end behavior must be considered on both the right and left.<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n","protected":false},"author":158108,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1930","chapter","type-chapter","status-publish","hentry"],"part":1198,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1930","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/158108"}],"version-history":[{"count":22,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1930\/revisions"}],"predecessor-version":[{"id":3251,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1930\/revisions\/3251"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/1198"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1930\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=1930"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1930"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=1930"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=1930"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}