{"id":4944,"date":"2017-12-29T15:17:07","date_gmt":"2017-12-29T15:17:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/?post_type=chapter&#038;p=4944"},"modified":"2018-05-17T00:58:02","modified_gmt":"2018-05-17T00:58:02","slug":"rational-expressions-and-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/rational-expressions-and-functions\/","title":{"raw":"Rational Expressions and Functions","rendered":"Rational Expressions and Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Recognize and define a rational expression<\/li>\r\n \t<li>Determine the domain of a rational expression<\/li>\r\n \t<li>Simplify a rational expression<\/li>\r\n \t<li>Rational functions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<strong>Rational expressions<\/strong> are fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem complicated because they contain variables, they can be simplified\u00a0using the techniques used to simplify expressions such as [latex]\\frac{4x^3}{12x^2}[\/latex] combined with techniques for factoring polynomials. There are a couple ways to get yourself into trouble when working with rational expressions, equations and functions. \u00a0One of them is dividing by zero, and the other is trying to divide across addition or subtraction.\r\n<h2>Determine the domain of a rational expression<\/h2>\r\nOne sure way you can break math is to divide by zero. Consider the following rational expression evaluated at x = 2:\r\n<p style=\"text-align: center\">Evaluate \u00a0[latex]\\displaystyle \\frac{x}{x-2}[\/latex] for [latex]x=2[\/latex]<\/p>\r\n<p style=\"text-align: center\">Substitute [latex]x=2[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}\\frac{2}{2-2}\\\\\\text{}\\\\=\\frac{2}{0}\\end{array}[\/latex]<\/p>\r\nThis means that for the expression [latex]\\displaystyle \\frac{x}{x-2}[\/latex], x cannot be 2 because it will result in an undefined ratio. In general, finding values for a variable that will not result in division by zero is called finding the domain. Finding the domain of a rational expression or function will help you not break math.\r\n<div class=\"textbox shaded\">\r\n<h4>Domain of a rational expression or equation<\/h4>\r\nThe domain of a rational expression or equation\u00a0is a collection of the values for the variable that will not result in an undefined mathematical operation such as division by zero. \u00a0For a = any real number, we can notate the domain in the following way:\r\n<p style=\"text-align: center\">x is all real numbers where [latex]x\\neq{a}[\/latex]<\/p>\r\n\r\n<\/div>\r\nThe reason you cannot divide any number <i>c<\/i> by zero [latex]\\displaystyle \\left( \\frac{c}{0}\\,\\,=\\,\\,? \\right)\\\\[\/latex] is that you would have to find a number that when you multiply it by 0 you would get back [latex]c \\left( ?\\,\\,\\cdot \\,\\,0\\,\\,=\\,\\,c \\right)[\/latex]. There are no numbers that can do this, so we say \u201cdivision by zero is undefined\u201d. In simplifying rational expressions you need to pay attention to what values of the variable(s) in the expression would make the denominator equal zero. These values cannot be included in the domain, so they're called excluded values. Discard them right at the start, before you go any further.\r\n\r\n(Note that although the <i>denominator<\/i> cannot be equivalent to 0, the <i>numerator<\/i> can\u2014this is why you only look for excluded values in the denominator of a rational expression.)\r\n\r\nFor rational expressions, the domain will exclude values for which the value of the denominator is 0. The following example illustrates finding the domain of an expression. Note that this is exactly the same algebra used to find the domain of a function.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIdentify the domain of the expression.\u00a0[latex]\\displaystyle \\frac{x+7}{{{x}^{2}}+8x-9}[\/latex]\r\n[reveal-answer q=\"318517\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"318517\"]\r\n\r\nFind any values for <i>x <\/i>that would make the denominator equal to 0 by setting the denominator equal to 0 and solving the equation.\r\n<p style=\"text-align: center\">[latex]x^{2}+8x-9=0[\/latex]<\/p>\r\nSolve the equation by factoring. The solutions are the values that are excluded from the domain.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}(x+9)(x-1)=0\\\\x=-9\\,\\,\\,\\text{or}\\,\\,\\,x=1\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe domain is all real numbers except [latex]\u22129[\/latex] and [latex]1[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Simplify Rational Expressions<\/h2>\r\nBefore we dive in to\u00a0simplifying rational expressions, let's review the difference between a factor, \u00a0a term, \u00a0and an expression. \u00a0This will hopefully help you avoid another way to break math\u00a0when you are simplifying rational expressions.\r\n\r\n<strong>Factors<\/strong> are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: 2 and 10 are factors of 20, as are 4, 5, 1, 20.\r\n\r\n<strong>Terms\u00a0<\/strong>are single numbers, or variables and numbers connected by multiplication. -4, 6x and [latex]x^2[\/latex] are all terms.\r\n\r\n<strong>Expressions <\/strong>are<strong>\u00a0<\/strong>groups of terms connected by addition and subtraction.\u00a0 [latex]2x^2-5[\/latex] is an expression.\r\n\r\nThis distinction is important when you are required to divide. \u00a0Let's use an example to show why this is important.\r\n\r\nSimplify: [latex]\\displaystyle \\large\\frac{2x^2}{12x}[\/latex]\r\n\r\nThe numerator and denominator of this fraction consist of factors. To simplify it, we can divide without being impeded by addition or subtraction.\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{cc}\\Large\\frac{2x^2}{12x}\\\\=\\Large\\frac{2\\cdot{x}\\cdot{x}}{2\\cdot3\\cdot2\\cdot{x}}\\\\=\\Large\\frac{\\cancel{2}\\cdot{\\cancel{x}}\\cdot{x}}{\\cancel{2}\\cdot3\\cdot2\\cdot{\\cancel{x}}}\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">We can do this because [latex]\\displaystyle \\frac{2}{2}=1\\text{ and }\\frac{x}{x}=1[\/latex], so our expression simplifies to [latex]\\displaystyle \\large\\frac{x}{6}[\/latex]<\/p>\r\nCompare that to\u00a0the expression [latex]\\displaystyle \\large\\frac{2x^2+x}{12-2x}[\/latex], notice the denominator and numerator consist of two terms connected by addition and subtraction. \u00a0We have to tip-toe around the addition and subtraction. \u00a0When asked to simplify it is tempting to want to cancel out like terms as we did when we just had factors. But you can't do that, it will break math!\r\n\r\n[caption id=\"attachment_2989\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-2989\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/22183401\/Screen-Shot-2016-07-22-at-11.32.38-AM-300x199.png\" alt=\"Shattered pottery strewn across the floor.\" width=\"300\" height=\"199\" \/> Breaking Math[\/caption]\r\n\r\nIn the examples that follow, the numerator and the denominator are polynomials with more than one term, and we will show you how to properly simplify them by factoring - which turns expressions connected by addition and subtraction into terms connected by multiplication.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify and state the domain for the expression.\u00a0[latex]\\displaystyle \\frac{x+3}{{{x}^{2}}+12x+27}[\/latex]\r\n[reveal-answer q=\"623785\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"623785\"]\r\n\r\nTo find the domain (and the excluded values), find the values for which the denominator is equal to 0. Factor the quadratic, and apply the zero product principle.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}x+3=0\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x+9=0\\\\x=0-3\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x=0-9\\\\x=-3\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x=-9\\\\\\\\x=-3\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x=-9\\end{array}[\/latex]<\/p>\r\nThe domain is all real numbers except [latex]x=-3[\/latex] or [latex]x=-9[\/latex].\r\n\r\nFactor the numerator and denominator. \u00a0Identify the factors that are the same in the numerator and denominator, and simplify.\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\large\\begin{array}{c}\\frac{x+3}{x^{2}+12x+27}\\\\\\\\=\\frac{x+3}{\\left(x+3\\right)\\left(x+9\\right)}\\\\\\\\\\frac{\\cancel{x+3}}{\\cancel{\\left(x+3\\right)}\\left(x+9\\right)}\\\\\\\\\\normalsize=1\\cdot\\large\\frac{1}{x+9}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\displaystyle\u00a0 \\frac{x+3}{{{x}^{2}}+12x+27}=\\frac{1}{x+9}[\/latex]\r\n\r\nThe domain is all real numbers except [latex]\u22123[\/latex] and [latex]\u22129[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify and state the domain for the expression.\u00a0[latex]\\displaystyle \\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}[\/latex]\r\n[reveal-answer q=\"861958\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"861958\"]\r\n\r\nTo find the domain, determine the values for which the denominator is equal to 0.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}x^{3}-x^{2}-20x=0\\\\x\\left(x^{2}-x-20\\right)=0\\\\x\\left(x-5\\right)\\left(x+4\\right)=0\\end{array}[\/latex]<\/p>\r\nThe domain is all real numbers except 0, 5, and \u22124.\r\n\r\nTo simplify, factor the numerator and denominator of the rational expression. Identify the factors that are the same in the numerator and denominator, and simplify.\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\large\\begin{array}{c}\\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}\\\\\\\\=\\frac{\\left(x+4\\right)\\left(x+6\\right)}{x\\left(x-5\\right)\\left(x+4\\right)}\\\\\\\\=\\frac{\\cancel{\\left(x+4\\right)}\\left(x+6\\right)}{x\\left(x-5\\right)\\cancel{\\left(x+4\\right)}}\\end{array}[\/latex]<\/p>\r\nSimplify. It is acceptable to either leave the denominator in factored form or to distribute multiplication.\r\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{x+6}{x\\left(x-5\\right)}\\,\\,\\,\\text{or}\\,\\,\\,\\frac{x+6}{x^{2}-5x}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle \\frac{x+6}{x(x-5)}[\/latex] or [latex] \\frac{x+6}{{{x}^{2}}-5x}[\/latex]\r\n\r\nThe domain is all real numbers except 0, 5, and [latex]\u22124[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe will show one last example of simplifying a rational expression. See if you can recognize the special product in the numerator.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]\\displaystyle \\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex], state the domain.\r\n[reveal-answer q=\"773059\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"773059\"]\r\n\r\nThe special product in the numerator is a difference of squares.\r\n\r\n[latex]\\displaystyle \\begin{array}\\frac{\\left(x+3\\right)\\left(x - 3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Factor the numerator and the denominator}.\\hfill \\\\ \\frac{x - 3}{x+1}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Cancel common factor }\\left(x+3\\right).\\hfill \\end{array}[\/latex]\r\n\r\nWith the denominator factored it is easier to find the domain of the expression. Determine the values for which the denominator is equal to 0.\r\n\r\n[latex]\\begin{array}{cc}\\left(x+3\\right)=0,\\left(x+1\\right)=0\\\\x\\ne-3,\\text{ AND }x\\ne-1\\end{array}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\displaystyle \\frac{{x}^{2}-9}{{x}^{2}+4x+3}=\\frac{x - 3}{x+1}[\/latex], Domain: [latex]x\\ne-3,\\text{ AND }x\\ne-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the following video we present another example of finding the domain of a rational expression.\r\nhttps:\/\/youtu.be\/tJiz5rEktBs\r\n<div class=\"textbox shaded\">\r\n<h3>Steps for Simplifying a Rational Expression<\/h3>\r\nTo simplify a rational expression, follow these steps:\r\n<ul>\r\n \t<li>Determine the domain. The excluded values are those values for the variable that result in the expression having a denominator of 0.<\/li>\r\n \t<li>Factor the numerator and denominator.<\/li>\r\n \t<li>Find common factors for the numerator and denominator and simplify.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Rational Functions<\/h2>\r\nWe started this section stating that a\u00a0rational expression\u00a0is an expression of the form [latex]\\displaystyle \\frac{p}{q} [\/latex] where [latex]p[\/latex]\u00a0and\u00a0<i>\u00a0<\/i>[latex]q[\/latex]\u00a0are polynomials and [latex]q(x) \\neq 0.[\/latex] Similarly, we define a\u00a0rational function\u00a0as a function of the form:\r\n<p style=\"text-align: center\">[latex]\\displaystyle R(x) = \\frac{p(x)}{q(x)}[\/latex]<\/p>\r\nwhere\u00a0[latex]p(x)[\/latex] and [latex]q(x)[\/latex]\u00a0are polynomial functions and\u00a0[latex]q(x)[\/latex] is not zero.\r\n\r\nThe domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make \u00a0[latex]q(x) = 0[\/latex].\r\n\r\nFor example,\u00a0[latex]\\displaystyle f(x) = \\frac{1}{x}[\/latex] and [latex]\\displaystyle f(x) = \\frac{1}{x^2}[\/latex] are examples of rational functions.\r\n<h3>Finding the domain of rational functions<\/h3>\r\nFor the rational function [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] (also called the reciprocal function), we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex]\\left\\{x|\\text{ }x\\ne 0\\right\\}[\/latex], the set of all real numbers that are not zero.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193614\/CNX_Precalc_Figure_01_02_0162.jpg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/>\r\n\r\nFor the rational function\u00a0[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex], we cannot divide by [latex]0[\/latex], so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193617\/CNX_Precalc_Figure_01_02_0172.jpg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/>\r\n<div class=\"textbox\">\r\n<h3>How To:\u00a0Given a rational function, find the domain.<\/h3>\r\n<ol>\r\n \t<li>Identify the input values.<\/li>\r\n \t<li>Identify any restrictions on the input. If there is a denominator in the function\u2019s formula, set the denominator equal to zero and solve for [latex]x[\/latex] . If the function\u2019s formula contains an even root, set the radicand greater than or equal to 0, and then solve.<\/li>\r\n \t<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Domain of a Rational Function<\/h3>\r\nFind the domain of the function [latex]\\displaystyle f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex].\r\n\r\n[reveal-answer q=\"759017\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"759017\"]\r\nWhen there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for [latex]x[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{cases}2-x=0\\hfill \\\\ -x=-2\\hfill \\\\ x=2\\hfill \\end{cases}[\/latex]<\/p>\r\nNow, we will exclude 2 from the domain. The answers are all real numbers where [latex]x&lt;2[\/latex] or [latex]x&gt;2[\/latex]. We can use a symbol known as the union, [latex]\\cup [\/latex], to combine the two sets. In interval notation, we write the solution: [latex]\\left(\\mathrm{-\\infty },2\\right)\\cup \\left(2,\\infty \\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193532\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/>\r\n\r\nIn interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,2\\right)\\cup \\left(2,\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/www.youtube.com\/watch?v=v0IhvIzCc_I&amp;feature=youtu.be\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the domain of the function: [latex]\\displaystyle f\\left(x\\right)=\\frac{1+4x}{2x - 1}[\/latex].\r\n\r\n[reveal-answer q=\"307426\"]Answer[\/reveal-answer]\r\n[hidden-answer a=\"307426\"]\r\n\r\n[latex]\\left(-\\infty ,\\frac{1}{2}\\right)\\cup \\left(\\frac{1}{2},\\infty \\right)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It!<\/h3>\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61836&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Recognize and define a rational expression<\/li>\n<li>Determine the domain of a rational expression<\/li>\n<li>Simplify a rational expression<\/li>\n<li>Rational functions<\/li>\n<\/ul>\n<\/div>\n<p><strong>Rational expressions<\/strong> are fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem complicated because they contain variables, they can be simplified\u00a0using the techniques used to simplify expressions such as [latex]\\frac{4x^3}{12x^2}[\/latex] combined with techniques for factoring polynomials. There are a couple ways to get yourself into trouble when working with rational expressions, equations and functions. \u00a0One of them is dividing by zero, and the other is trying to divide across addition or subtraction.<\/p>\n<h2>Determine the domain of a rational expression<\/h2>\n<p>One sure way you can break math is to divide by zero. Consider the following rational expression evaluated at x = 2:<\/p>\n<p style=\"text-align: center\">Evaluate \u00a0[latex]\\displaystyle \\frac{x}{x-2}[\/latex] for [latex]x=2[\/latex]<\/p>\n<p style=\"text-align: center\">Substitute [latex]x=2[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{l}\\frac{2}{2-2}\\\\\\text{}\\\\=\\frac{2}{0}\\end{array}[\/latex]<\/p>\n<p>This means that for the expression [latex]\\displaystyle \\frac{x}{x-2}[\/latex], x cannot be 2 because it will result in an undefined ratio. In general, finding values for a variable that will not result in division by zero is called finding the domain. Finding the domain of a rational expression or function will help you not break math.<\/p>\n<div class=\"textbox shaded\">\n<h4>Domain of a rational expression or equation<\/h4>\n<p>The domain of a rational expression or equation\u00a0is a collection of the values for the variable that will not result in an undefined mathematical operation such as division by zero. \u00a0For a = any real number, we can notate the domain in the following way:<\/p>\n<p style=\"text-align: center\">x is all real numbers where [latex]x\\neq{a}[\/latex]<\/p>\n<\/div>\n<p>The reason you cannot divide any number <i>c<\/i> by zero [latex]\\displaystyle \\left( \\frac{c}{0}\\,\\,=\\,\\,? \\right)\\\\[\/latex] is that you would have to find a number that when you multiply it by 0 you would get back [latex]c \\left( ?\\,\\,\\cdot \\,\\,0\\,\\,=\\,\\,c \\right)[\/latex]. There are no numbers that can do this, so we say \u201cdivision by zero is undefined\u201d. In simplifying rational expressions you need to pay attention to what values of the variable(s) in the expression would make the denominator equal zero. These values cannot be included in the domain, so they&#8217;re called excluded values. Discard them right at the start, before you go any further.<\/p>\n<p>(Note that although the <i>denominator<\/i> cannot be equivalent to 0, the <i>numerator<\/i> can\u2014this is why you only look for excluded values in the denominator of a rational expression.)<\/p>\n<p>For rational expressions, the domain will exclude values for which the value of the denominator is 0. The following example illustrates finding the domain of an expression. Note that this is exactly the same algebra used to find the domain of a function.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Identify the domain of the expression.\u00a0[latex]\\displaystyle \\frac{x+7}{{{x}^{2}}+8x-9}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q318517\">Show Solution<\/span><\/p>\n<div id=\"q318517\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find any values for <i>x <\/i>that would make the denominator equal to 0 by setting the denominator equal to 0 and solving the equation.<\/p>\n<p style=\"text-align: center\">[latex]x^{2}+8x-9=0[\/latex]<\/p>\n<p>Solve the equation by factoring. The solutions are the values that are excluded from the domain.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}(x+9)(x-1)=0\\\\x=-9\\,\\,\\,\\text{or}\\,\\,\\,x=1\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The domain is all real numbers except [latex]\u22129[\/latex] and [latex]1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Simplify Rational Expressions<\/h2>\n<p>Before we dive in to\u00a0simplifying rational expressions, let&#8217;s review the difference between a factor, \u00a0a term, \u00a0and an expression. \u00a0This will hopefully help you avoid another way to break math\u00a0when you are simplifying rational expressions.<\/p>\n<p><strong>Factors<\/strong> are the building blocks of multiplication. They are the numbers that you can multiply together to produce another number: 2 and 10 are factors of 20, as are 4, 5, 1, 20.<\/p>\n<p><strong>Terms\u00a0<\/strong>are single numbers, or variables and numbers connected by multiplication. -4, 6x and [latex]x^2[\/latex] are all terms.<\/p>\n<p><strong>Expressions <\/strong>are<strong>\u00a0<\/strong>groups of terms connected by addition and subtraction.\u00a0 [latex]2x^2-5[\/latex] is an expression.<\/p>\n<p>This distinction is important when you are required to divide. \u00a0Let&#8217;s use an example to show why this is important.<\/p>\n<p>Simplify: [latex]\\displaystyle \\large\\frac{2x^2}{12x}[\/latex]<\/p>\n<p>The numerator and denominator of this fraction consist of factors. To simplify it, we can divide without being impeded by addition or subtraction.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\begin{array}{cc}\\Large\\frac{2x^2}{12x}\\\\=\\Large\\frac{2\\cdot{x}\\cdot{x}}{2\\cdot3\\cdot2\\cdot{x}}\\\\=\\Large\\frac{\\cancel{2}\\cdot{\\cancel{x}}\\cdot{x}}{\\cancel{2}\\cdot3\\cdot2\\cdot{\\cancel{x}}}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">We can do this because [latex]\\displaystyle \\frac{2}{2}=1\\text{ and }\\frac{x}{x}=1[\/latex], so our expression simplifies to [latex]\\displaystyle \\large\\frac{x}{6}[\/latex]<\/p>\n<p>Compare that to\u00a0the expression [latex]\\displaystyle \\large\\frac{2x^2+x}{12-2x}[\/latex], notice the denominator and numerator consist of two terms connected by addition and subtraction. \u00a0We have to tip-toe around the addition and subtraction. \u00a0When asked to simplify it is tempting to want to cancel out like terms as we did when we just had factors. But you can&#8217;t do that, it will break math!<\/p>\n<div id=\"attachment_2989\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2989\" class=\"size-medium wp-image-2989\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/22183401\/Screen-Shot-2016-07-22-at-11.32.38-AM-300x199.png\" alt=\"Shattered pottery strewn across the floor.\" width=\"300\" height=\"199\" \/><\/p>\n<p id=\"caption-attachment-2989\" class=\"wp-caption-text\">Breaking Math<\/p>\n<\/div>\n<p>In the examples that follow, the numerator and the denominator are polynomials with more than one term, and we will show you how to properly simplify them by factoring &#8211; which turns expressions connected by addition and subtraction into terms connected by multiplication.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify and state the domain for the expression.\u00a0[latex]\\displaystyle \\frac{x+3}{{{x}^{2}}+12x+27}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q623785\">Show Solution<\/span><\/p>\n<div id=\"q623785\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find the domain (and the excluded values), find the values for which the denominator is equal to 0. Factor the quadratic, and apply the zero product principle.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}x+3=0\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x+9=0\\\\x=0-3\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x=0-9\\\\x=-3\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x=-9\\\\\\\\x=-3\\,\\,\\,\\,\\,\\,\\text{or}\\,\\,\\,\\,\\,\\,x=-9\\end{array}[\/latex]<\/p>\n<p>The domain is all real numbers except [latex]x=-3[\/latex] or [latex]x=-9[\/latex].<\/p>\n<p>Factor the numerator and denominator. \u00a0Identify the factors that are the same in the numerator and denominator, and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\large\\begin{array}{c}\\frac{x+3}{x^{2}+12x+27}\\\\\\\\=\\frac{x+3}{\\left(x+3\\right)\\left(x+9\\right)}\\\\\\\\\\frac{\\cancel{x+3}}{\\cancel{\\left(x+3\\right)}\\left(x+9\\right)}\\\\\\\\\\normalsize=1\\cdot\\large\\frac{1}{x+9}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle\u00a0 \\frac{x+3}{{{x}^{2}}+12x+27}=\\frac{1}{x+9}[\/latex]<\/p>\n<p>The domain is all real numbers except [latex]\u22123[\/latex] and [latex]\u22129[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify and state the domain for the expression.\u00a0[latex]\\displaystyle \\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q861958\">Show Solution<\/span><\/p>\n<div id=\"q861958\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find the domain, determine the values for which the denominator is equal to 0.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}x^{3}-x^{2}-20x=0\\\\x\\left(x^{2}-x-20\\right)=0\\\\x\\left(x-5\\right)\\left(x+4\\right)=0\\end{array}[\/latex]<\/p>\n<p>The domain is all real numbers except 0, 5, and \u22124.<\/p>\n<p>To simplify, factor the numerator and denominator of the rational expression. Identify the factors that are the same in the numerator and denominator, and simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\large\\begin{array}{c}\\frac{x^{2}+10x+24}{x^{3}-x^{2}-20x}\\\\\\\\=\\frac{\\left(x+4\\right)\\left(x+6\\right)}{x\\left(x-5\\right)\\left(x+4\\right)}\\\\\\\\=\\frac{\\cancel{\\left(x+4\\right)}\\left(x+6\\right)}{x\\left(x-5\\right)\\cancel{\\left(x+4\\right)}}\\end{array}[\/latex]<\/p>\n<p>Simplify. It is acceptable to either leave the denominator in factored form or to distribute multiplication.<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle \\frac{x+6}{x\\left(x-5\\right)}\\,\\,\\,\\text{or}\\,\\,\\,\\frac{x+6}{x^{2}-5x}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{x+6}{x(x-5)}[\/latex] or [latex]\\frac{x+6}{{{x}^{2}}-5x}[\/latex]<\/p>\n<p>The domain is all real numbers except 0, 5, and [latex]\u22124[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We will show one last example of simplifying a rational expression. See if you can recognize the special product in the numerator.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\displaystyle \\frac{{x}^{2}-9}{{x}^{2}+4x+3}[\/latex], state the domain.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q773059\">Show Answer<\/span><\/p>\n<div id=\"q773059\" class=\"hidden-answer\" style=\"display: none\">\n<p>The special product in the numerator is a difference of squares.<\/p>\n<p>[latex]\\displaystyle \\begin{array}\\frac{\\left(x+3\\right)\\left(x - 3\\right)}{\\left(x+3\\right)\\left(x+1\\right)}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Factor the numerator and the denominator}.\\hfill \\\\ \\frac{x - 3}{x+1}\\hfill & \\hfill & \\hfill & \\hfill & \\text{Cancel common factor }\\left(x+3\\right).\\hfill \\end{array}[\/latex]<\/p>\n<p>With the denominator factored it is easier to find the domain of the expression. Determine the values for which the denominator is equal to 0.<\/p>\n<p>[latex]\\begin{array}{cc}\\left(x+3\\right)=0,\\left(x+1\\right)=0\\\\x\\ne-3,\\text{ AND }x\\ne-1\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle \\frac{{x}^{2}-9}{{x}^{2}+4x+3}=\\frac{x - 3}{x+1}[\/latex], Domain: [latex]x\\ne-3,\\text{ AND }x\\ne-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the following video we present another example of finding the domain of a rational expression.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Simplify and Give the Domain of Rational Expressions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tJiz5rEktBs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox shaded\">\n<h3>Steps for Simplifying a Rational Expression<\/h3>\n<p>To simplify a rational expression, follow these steps:<\/p>\n<ul>\n<li>Determine the domain. The excluded values are those values for the variable that result in the expression having a denominator of 0.<\/li>\n<li>Factor the numerator and denominator.<\/li>\n<li>Find common factors for the numerator and denominator and simplify.<\/li>\n<\/ul>\n<\/div>\n<h2>Rational Functions<\/h2>\n<p>We started this section stating that a\u00a0rational expression\u00a0is an expression of the form [latex]\\displaystyle \\frac{p}{q}[\/latex] where [latex]p[\/latex]\u00a0and\u00a0<i>\u00a0<\/i>[latex]q[\/latex]\u00a0are polynomials and [latex]q(x) \\neq 0.[\/latex] Similarly, we define a\u00a0rational function\u00a0as a function of the form:<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle R(x) = \\frac{p(x)}{q(x)}[\/latex]<\/p>\n<p>where\u00a0[latex]p(x)[\/latex] and [latex]q(x)[\/latex]\u00a0are polynomial functions and\u00a0[latex]q(x)[\/latex] is not zero.<\/p>\n<p>The domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make \u00a0[latex]q(x) = 0[\/latex].<\/p>\n<p>For example,\u00a0[latex]\\displaystyle f(x) = \\frac{1}{x}[\/latex] and [latex]\\displaystyle f(x) = \\frac{1}{x^2}[\/latex] are examples of rational functions.<\/p>\n<h3>Finding the domain of rational functions<\/h3>\n<p>For the rational function [latex]f\\left(x\\right)=\\frac{1}{x}[\/latex] (also called the reciprocal function), we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write [latex]\\left\\{x|\\text{ }x\\ne 0\\right\\}[\/latex], the set of all real numbers that are not zero.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193614\/CNX_Precalc_Figure_01_02_0162.jpg\" alt=\"Reciprocal function f(x)=1\/x.\" width=\"487\" height=\"433\" \/><\/p>\n<p>For the rational function\u00a0[latex]f\\left(x\\right)=\\frac{1}{{x}^{2}}[\/latex], we cannot divide by [latex]0[\/latex], so we must exclude [latex]0[\/latex] from the domain. There is also no [latex]x[\/latex] that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193617\/CNX_Precalc_Figure_01_02_0172.jpg\" alt=\"Reciprocal squared function f(x)=1\/x^2\" width=\"487\" height=\"433\" \/><\/p>\n<div class=\"textbox\">\n<h3>How To:\u00a0Given a rational function, find the domain.<\/h3>\n<ol>\n<li>Identify the input values.<\/li>\n<li>Identify any restrictions on the input. If there is a denominator in the function\u2019s formula, set the denominator equal to zero and solve for [latex]x[\/latex] . If the function\u2019s formula contains an even root, set the radicand greater than or equal to 0, and then solve.<\/li>\n<li>Write the domain in interval form, making sure to exclude any restricted values from the domain.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Domain of a Rational Function<\/h3>\n<p>Find the domain of the function [latex]\\displaystyle f\\left(x\\right)=\\frac{x+1}{2-x}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q759017\">Solution<\/span><\/p>\n<div id=\"q759017\" class=\"hidden-answer\" style=\"display: none\">\nWhen there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for [latex]x[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{cases}2-x=0\\hfill \\\\ -x=-2\\hfill \\\\ x=2\\hfill \\end{cases}[\/latex]<\/p>\n<p>Now, we will exclude 2 from the domain. The answers are all real numbers where [latex]x<2[\/latex] or [latex]x>2[\/latex]. We can use a symbol known as the union, [latex]\\cup[\/latex], to combine the two sets. In interval notation, we write the solution: [latex]\\left(\\mathrm{-\\infty },2\\right)\\cup \\left(2,\\infty \\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/18193532\/CNX_Precalc_Figure_01_02_028n2.jpg\" alt=\"Line graph of x=!2.\" width=\"487\" height=\"164\" \/><\/p>\n<p>In interval form, the domain of [latex]f[\/latex] is [latex]\\left(-\\infty ,2\\right)\\cup \\left(2,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  The Domain of Rational Functions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/v0IhvIzCc_I?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the domain of the function: [latex]\\displaystyle f\\left(x\\right)=\\frac{1+4x}{2x - 1}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q307426\">Answer<\/span><\/p>\n<div id=\"q307426\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\infty ,\\frac{1}{2}\\right)\\cup \\left(\\frac{1}{2},\\infty \\right)[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It!<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=61836&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4944\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 15: Rational Expressions, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Simplify and Give the Domain of Rational Expressions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/tJiz5rEktBs\">https:\/\/youtu.be\/tJiz5rEktBs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at :http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/li><li>Question ID#61836. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":60342,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Unit 15: Rational Expressions, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Simplify and Give the Domain of Rational 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