{"id":6636,"date":"2020-10-03T15:16:15","date_gmt":"2020-10-03T15:16:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/slcc-beginalgebra\/?post_type=chapter&p=6636"},"modified":"2026-01-17T05:24:13","modified_gmt":"2026-01-17T05:24:13","slug":"3-1-intro-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-elementaryalgebra\/chapter\/3-1-intro-functions\/","title":{"raw":"3.2: Intro to Functions","rendered":"3.2: Intro to Functions"},"content":{"raw":"
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section 3.2 Learning Objectives<\/h3>\r\n3.2: Introduction to Functions<\/strong>\r\n
    \r\n \t
  • Determine if a relation is a function<\/li>\r\n \t
  • Identify the domain and range of a set of ordered pairs<\/li>\r\n \t
  • Evaluate a function written in function notation<\/li>\r\n \t
  • Find the domain of a rational function<\/li>\r\n<\/ul>\r\n<\/div>\r\n \r\n\r\nThere are many kinds of relations. A relation is simply a correspondence between sets of values or information. Think about members of your family and their ages. The pairing of each member of your family and their age is a relation. Each family member can be paired with an age in the set of ages of your family members. Another example of a relation is the pairing of a state with its U.S. senators. Each state can be matched with two individuals who have each been elected to serve as a senator. In turn, each senator can be matched with one specific state that he or she represents. Both of these are real-life examples of relations.\r\n

    Determine if a relation is a function<\/h2>\r\nThe first value of a relation is an input value and the second value is the output value. A function<\/strong> is a specific type of relation in which each input value has one and only one output value. An input is the independent<\/i> value, and the output is the dependent <\/i>value, as it depends on the value of the input.\r\n\r\nWhich set of values make up the input values and output values can affect whether or not the relation is a function. Notice in the first table below, where the input is \u201cname\u201d and the output is \u201cage,\u201d each input matches with exactly one output. This is an example of a function.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Family Member's Name (Input)<\/th>\r\nFamily Member's Age (Output)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    Nellie<\/td>\r\n[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n
    Marcos<\/td>\r\n[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n
    Esther<\/td>\r\n[latex]46[\/latex]<\/td>\r\n<\/tr>\r\n
    Samuel<\/td>\r\n[latex]47[\/latex]<\/td>\r\n<\/tr>\r\n
    Nina<\/td>\r\n[latex]47[\/latex]<\/td>\r\n<\/tr>\r\n
    Paul<\/td>\r\n[latex]47[\/latex]<\/td>\r\n<\/tr>\r\n
    Katrina<\/td>\r\n[latex]21[\/latex]<\/td>\r\n<\/tr>\r\n
    Andrew<\/td>\r\n[latex]16[\/latex]<\/td>\r\n<\/tr>\r\n
    Maria<\/td>\r\n[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n
    Ana<\/td>\r\n[latex]81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCompare this with the next table where the input is \u201cage\u201d and the output is \u201cname.\u201d Some of the inputs result in more than one output. This is an example of a correspondence that is not <\/i>a function.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Family Member's Age (Input)<\/th>\r\nFamily Member's Name (Output)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    [latex]11[\/latex]<\/td>\r\nMarcos<\/td>\r\n<\/tr>\r\n
    [latex]13[\/latex]<\/td>\r\nNellie, Maria<\/td>\r\n<\/tr>\r\n
    [latex]16[\/latex]<\/td>\r\nAndrew<\/td>\r\n<\/tr>\r\n
    [latex]21[\/latex]<\/td>\r\nKatrina<\/td>\r\n<\/tr>\r\n
    [latex]46[\/latex]<\/td>\r\nEsther<\/td>\r\n<\/tr>\r\n
    [latex]47[\/latex]<\/td>\r\nSamuel, Nina, Paul<\/td>\r\n<\/tr>\r\n
    [latex]81[\/latex]<\/td>\r\nAna<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow let us look at some other examples to determine whether the relations are functions or not and under what circumstances. Remember that a relation is a function if there is only one <\/strong><\/em>output for each input.\r\n
    \r\n

    Example 1<\/h3>\r\nFill in the table.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Input<\/th>\r\nOutput<\/th>\r\nFunction?<\/th>\r\nWhy or why not?<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    Name of senator<\/td>\r\nName of state<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    Name of state<\/td>\r\nName of senator<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    Time elapsed<\/td>\r\nHeight of a tossed ball<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    Height of a tossed ball<\/td>\r\nTime elapsed<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    Number of cars<\/td>\r\nNumber of tires<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
    Number of tires<\/td>\r\nNumber of cars<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"842346\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"842346\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Input<\/th>\r\nOutput<\/th>\r\nFunction?<\/th>\r\nWhy or why not?<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    Name of senator<\/td>\r\nName of state<\/td>\r\nYes<\/td>\r\nFor each input, there will only be one output because a senator only represents one state.<\/td>\r\n<\/tr>\r\n
    Name of state<\/td>\r\nName of senator<\/td>\r\nNo<\/td>\r\nFor each state that is an input, 2 names of senators would result because each state has two senators.<\/td>\r\n<\/tr>\r\n
    Time elapsed<\/td>\r\nHeight of a tossed ball<\/td>\r\nYes<\/td>\r\nAt a specific time, the ball has one specific height.<\/td>\r\n<\/tr>\r\n
    Height of a tossed ball<\/td>\r\nTime elapsed<\/td>\r\nNo<\/td>\r\nRemember that the ball was tossed up and fell down. So for a given height, there could be two different times when the ball was at that height. The input height can result in more than one output.<\/td>\r\n<\/tr>\r\n
    Number of cars<\/td>\r\nNumber of tires<\/td>\r\nYes<\/td>\r\nFor any input of a specific number of cars, there is one specific output representing the number of tires.<\/td>\r\n<\/tr>\r\n
    Number of tires<\/td>\r\nNumber of cars<\/td>\r\nYes<\/td>\r\nFor any input of a specific number of tires, there is one specific output representing the number of cars (assuming each car has all four of its tires).<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nRelations can be written as ordered pairs of numbers or as numbers in a table of values. Unless stated otherwise, the first number of an ordered pair is the x<\/em>-value and is the input, while the second number is the\u00a0y<\/em>-value and is the output.\u00a0\u00a0By examining the inputs (x<\/i>-coordinates) and outputs (y<\/i>-coordinates), you can determine whether or not the relation is a function. Remember, in a function, each input has only one output.\r\n

    Identify the domain and range of a set of ordered pairs<\/h2>\r\nThere is a name for the set of input values and another name for the set of output values for a function. The set of input values is called the domain of the function<\/b>. The set of output values is called the range of the function<\/b>.\r\n\r\nIf you have a set of ordered pairs, you can find the domain by listing all of the input values, which are the x<\/i>-coordinates. To find the range, list all of the output values, which are the y<\/i>-coordinates.\r\n\r\nConsider the following set of ordered pairs:\r\n\r\n[latex]\\{(\u22122,0),(0,6),(2,12),(4,18)\\}[\/latex]\r\n\r\nYou have the following:\r\n\r\n[latex]\\begin{array}{l}\\text{Domain:}\\{\u22122,0,2,4\\}\\\\\\text{Range:}\\{0,6,12,18\\}\\end{array}[\/latex]\r\n\r\nNow try it yourself.\r\n
    \r\n

    Example 2<\/h3>\r\nList the domain and range for the following table of values where x<\/em> is the input and y<\/em> is the output. Then determine if the relation is a function.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    x<\/i><\/th>\r\ny<\/i><\/th>\r\n<\/tr>\r\n
    [latex]\u22123[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\u22122[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\u22121[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]2[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]3[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"594198\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"594198\"]\r\n\r\nThe domain describes all the inputs, and we can use set notation with brackets { } to make the list.\r\n\r\n[latex]\\text{Domain}:\\{-3,-2,-1,2,3\\}[\/latex]\r\n\r\nThe range describes all the outputs.\r\n\r\n[latex]\\text{Range}:\\{4\\}[\/latex]\r\n\r\nWe only listed\u00a0[latex]4[\/latex] once because it is not necessary to list a number every time it appears in a set.\r\n\r\nEach input has only one output, and the fact that it is the same output (4) does not matter.\r\n\r\nTherefore, this relation is a function.\r\n\r\nA helpful way to visualize this is through an \"arrow diagram.\"\u00a0 We list our domain and range (and as mentioned above with our range, do not list an element more than once in either set). Then, for each input in the range, we draw an arrow to the output it maps to in the range. This table of values corresponds to the arrow diagram shown below.\r\n

    \"Mapping<\/p>\r\nSince each input maps to only one output (equivalently, there is only one arrow coming from each input), the relation is a function.\r\n

    Answer<\/span><\/h4>\r\n[latex]\\text{Domain}:\\{-3,-2,-1,2,3\\}[\/latex]\r\n\r\n[latex]\\text{Range}:\\{4\\}[\/latex]\r\n\r\nThe relation is a function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we provide another example of identifying whether a table of values represents a function as well as determining the domain and range of each.\r\n\r\n[embed]https:\/\/youtu.be\/y2TqnP_6M1s[\/embed]\r\n
    \r\n

    Example 3<\/h3>\r\nDefine the domain and range for the following set of ordered pairs, and determine whether the relation given is a function.\r\n

    [latex]\\{(\u22123,\u22126),(\u22122,\u22121),(1,0),(1,5),(2,0)\\}[\/latex]<\/p>\r\n[reveal-answer q=\"507050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"507050\"]\r\n\r\nWe list all of the input values as the domain. \u00a0The input values are represented first in the ordered pair as a matter of convention.\r\n\r\nDomain: {[latex]-3,-2,1,2[\/latex]}\r\n\r\nNote how we did not enter repeated values more than once; it is not necessary.\r\n\r\nThe range is the list of outputs for the relation; they are entered second in the ordered pair.\r\n\r\nRange: {[latex]-6, -1, 0, 5[\/latex]}\r\n\r\nOrganizing the ordered pairs in a table can help you tell whether this relation is a function. \u00a0By definition, the inputs in a function have only one output.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    x<\/i><\/th>\r\ny<\/i><\/th>\r\n<\/tr>\r\n
    [latex]\u22123[\/latex]<\/td>\r\n[latex]\u22126[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]\u22122[\/latex]<\/td>\r\n[latex]\u22121[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]1[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]1[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]2[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe relation is not a function because the input\u00a0[latex]1[\/latex] has two outputs:\u00a0[latex]0[\/latex] and\u00a0[latex]5[\/latex].\r\n\r\nAgain, we could use an arrow diagram to see this visually. Making sure to only list the input 1 once in our domain, we get the following diagram:\r\n

    \"Mapping<\/p>\r\nBecause the input of 1 maps to two different outputs (two arrows coming from one input), this confirms that the relation is not a function.\r\n

    Answer<\/span><\/h4>\r\nDomain: {[latex]-3,-2,1,2[\/latex]}\r\n\r\nRange: {[latex]-6, -1, 0, 5[\/latex]}\r\n\r\nThe relation is not a function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show how to determine whether a relation is a function and how to find the domain and range.\r\n\r\n[embed]https:\/\/youtu.be\/kzgLfwgxE8g[\/embed]\r\n

    Summary: Determining Whether a Relation is a Function<\/strong><\/h3>\r\n
      \r\n \t
    1. Identify the input values - this is your domain.<\/li>\r\n \t
    2. Identify the output values - this is your range.<\/li>\r\n \t
    3. If each value in the domain leads to only one value in the range, classify the relationship as a function. If any value in the domain leads to two or more values in the range, do not classify the relationship as a function.<\/li>\r\n<\/ol>\r\n

      Function notation<\/h2>\r\nSome people think of functions as \u201cmathematical machines.\u201d Imagine you have a machine that changes a number according to a specific rule such as \u201cmultiply by\u00a0[latex]3[\/latex] and add\u00a0[latex]2[\/latex]\u201d or \u201cdivide by\u00a0[latex]5[\/latex], add\u00a0[latex]25[\/latex], and multiply by [latex]\u22121[\/latex].\u201d If you put a number into the machine, a new number will pop out the other end having been changed according to the rule. The number that goes in is called the input, and the number that is produced is called the output.\r\n\r\nYou can also call the machine \u201cf\u201d <\/i>for function. If you put x <\/i>into the machine, f<\/i>(x<\/i>), <\/i>comes out. Mathematically speaking, x<\/i> is the input, or the \u201cindependent variable,\u201d and f<\/i>(x<\/i>) is the output, or the \u201cdependent variable,\u201d since it depends on the value of x<\/i>.\r\n\r\n[latex]f(x)=4x+1[\/latex] is written in function notation and is read \u201cf<\/i> of x<\/i> equals\u00a0[latex]4x[\/latex] plus\u00a01\u201d It represents the following situation: A function named f <\/i>acts upon an input, x, <\/i>and produces f<\/i>(x<\/i>) which is equal to [latex]4x+1[\/latex]. This is the same as the equation [latex]y=4x+1[\/latex].\r\n\r\nFunction notation gives you more flexibility because you do not have to use [latex]y[\/latex] for every equation. Instead, you could use [latex]f(x)[\/latex] or [latex]g(x)[\/latex]\u00a0or even [latex]c(x)[\/latex]. This can be a helpful way to distinguish equations of functions when you are dealing with more than one at a time.\r\n

      Using Function Notation<\/h3>\r\nOnce we determine that a relationship is a function, we need to display and define the functional relationship so that we can understand it, use it, and possibly even program it into a computer. There are various ways of representing functions. A standard function notation<\/strong> is one representation that facilitates working with functions.\r\n
      \r\n

      A General Note: Function Notation<\/h3>\r\nThe notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as \"[latex]y[\/latex] is a function of [latex]x[\/latex].\" The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y[\/latex] <\/em>or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.\r\n\r\n<\/div>\r\n
      \r\n

      Example 4<\/h3>\r\nUse function notation to represent a function whose input is the name of a month and output is the number of days in that month.\r\n\r\n[reveal-answer q=\"5489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"5489\"]\r\n\r\nThe number of days in a month is a function of the name of the month, so if we name the function [latex]f[\/latex], we write [latex]f\\left(\\text{month}\\right)=\\text{days}[\/latex]\u00a0or [latex]f\\left(m\\right)=d[\/latex]. The name of the month is the input to a \"rule\" that associates a specific number (the output) with each input.\r\n\r\n\"The\r\n\r\nFor example, [latex]f\\left(\\text{March}\\right)=31[\/latex], because March has\u00a0[latex]31[\/latex] days. The notation [latex]d=f\\left(m\\right)[\/latex] reminds us that the number of days, [latex]d[\/latex] (the output), is dependent on the name of the month, [latex]m[\/latex] (the input).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNote that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with here will have numbers as inputs and outputs.\r\n
      \r\n

      Example 5<\/h3>\r\nA function [latex]f\\left(y\\right)=N[\/latex] gives the number of police officers, [latex]N[\/latex], in a town in year [latex]y[\/latex]. What does [latex]f\\left(2005\\right)=300[\/latex] represent?\r\n\r\n[reveal-answer q=\"226737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"226737\"]\r\n\r\nWhen we read [latex]f\\left(2005\\right)=300[\/latex], we see that the input year is 2005. The value for the output, the number of police officers, [latex]N[\/latex], is 300. The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were\u00a0[latex]300[\/latex] police officers in the town.\u00a0The notation [latex]N=f\\left(y\\right)[\/latex] reminds us that the number of police officers, [latex]N[\/latex] (the output), is dependent on the year, [latex]y[\/latex] (the input).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following videos we show two more\u00a0examples of how to express a relationship using function notation.\r\n\r\nhttps:\/\/youtu.be\/lF0fzdaxU_8\r\n\r\nhttps:\/\/youtu.be\/nAF_GZFwU1g\r\n

      Evaluating a function written in function notation<\/h2>\r\nThroughout this course, you have been and will continue working with algebraic equations. Many of these equations are functions. For example, [latex]y=4x+1[\/latex] is an equation that represents a function. When you input values for x<\/em>, you can determine a single output for y<\/em>. In this case, if you substitute [latex]x=10[\/latex] into the equation you will find that y must be\u00a0[latex]41[\/latex]; there is no other value of y that would make the equation true.\r\n\r\nRather than using the variable y, the equations of functions can be written using function notation<\/strong>. Function notation is very useful when you are working with more than one function at a time and substituting more than one value in for x<\/em>.\r\n\r\nEquations written using function notation can also be evaluated. With function notation, you might see the following:\r\n\r\nGiven [latex]f(x)=4x+1[\/latex], <\/i>find\u00a0[latex]f(2)[\/latex].\r\n\r\nYou read this problem like this: \u201cgiven [latex]f[\/latex] of [latex]x[\/latex]\u00a0equals\u00a0[latex]4x[\/latex] plus one, find [latex]f[\/latex] of\u00a0[latex]2[\/latex].\u201d While the notation and wording is different, the process of evaluating a function is the same as evaluating an expression at a specific value. In both cases, you substitute\u00a0[latex]2[\/latex] for [latex]x[\/latex], multiply it by\u00a0[latex]4[\/latex] and add\u00a0[latex]1[\/latex], simplifying to get\u00a0[latex]9[\/latex]. In this function, an input of\u00a0[latex]2[\/latex] results in an output of\u00a0[latex]9[\/latex].\r\n

      [latex]f(x)=4x+1\\\\f(2)=4(2)+1=8+1=9[\/latex]<\/p>\r\nYou can simply apply what you already know about evaluating expressions to evaluate a function. It is important to note that the parentheses that are part of function notation do not mean multiply. The notation [latex]f(x)[\/latex]\u00a0does not mean [latex]f[\/latex]\u00a0multiplied by [latex]x[\/latex]. Instead, the notation means \u201c[latex]f[\/latex]of [latex]x[\/latex]\u201d or \u201cthe function of [latex]x[\/latex].\"<\/i>\u00a0To evaluate the function, take the value given for [latex]x[\/latex],<\/i>\u00a0and substitute that value in for [latex]x[\/latex]<\/i> in the expression. Let us look at a couple of examples.\r\n

      \r\n

      Example 6<\/h3>\r\nGiven [latex]f(x)=3x\u20134[\/latex],\u00a0find [latex]f(5)[\/latex].\r\n\r\n[reveal-answer q=\"42679\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"42679\"]\r\n\r\nSubstitute\u00a0[latex]5[\/latex] in for x <\/i>in the function.\r\n

      [latex]f(5)=3(5)-4[\/latex]<\/p>\r\nSimplify the expression on the right side of the equation.\r\n

      [latex]f(5)=15-4[\/latex]<\/p>\r\n

      [latex]f(5)=11[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nFunctions can be evaluated for negative values of x<\/i>, too. Keep in mind the rules for integer operations.\r\n

      \r\n

      Example 7<\/h3>\r\nGiven [latex]p(x)=2x^{2}+5[\/latex], find [latex]p(\u22123)[\/latex].\r\n\r\n[reveal-answer q=\"489384\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"489384\"]\r\n\r\nSubstitute [latex]-3[\/latex] in for x <\/i>in the function.\r\n

      [latex]p(\u22123)=2(\u22123)^{2}+5[\/latex]<\/p>\r\nSimplify the expression on the right side of the equation.\r\n

      [latex]p(\u22123)=2(9)+5[\/latex]<\/p>\r\n

      [latex]p(\u22123)=18+5[\/latex]<\/p>\r\n

      [latex]p(\u22123)=23[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou may also be asked to evaluate a function for more than one value as shown in the example that follows.\r\n

      \r\n

      Example 8<\/h3>\r\nGiven [latex]f(x)=|4x-3|[\/latex], find [latex]f(0)[\/latex], [latex]f(2)[\/latex], and [latex]f(\u22121)[\/latex].\r\n\r\n[reveal-answer q=\"971051\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"971051\"]\r\n\r\nTreat each of these like three separate problems. In each case, you substitute the value in for x<\/em> and simplify.\r\n\r\nStart with [latex]x=0[\/latex].\r\n

      [latex]f(0)=|4(0)-3|=|-3|=3[\/latex]<\/p>\r\n

      [latex]f(0)=3[\/latex]<\/p>\r\nEvaluate for [latex]x=2[\/latex].\r\n

      [latex]f(2)=|4(2)-3|=|5|=5[\/latex]<\/p>\r\n

      [latex]f(2)=5[\/latex]<\/p>\r\nEvaluate for [latex]x=\u22121[\/latex].\r\n

      [latex]f(\u22121)=|4(-1)-3|=|-7|=7[\/latex]<\/p>\r\n

      [latex]f(-1)=7[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNext we look at an example where one of the inputs leads to a problem.\r\n

      \r\n

      Example 9<\/h3>\r\nGiven [latex]f(x)=\\displaystyle{ \\frac{2x}{x+4}}[\/latex], find [latex]f(0)[\/latex], [latex]f(3)[\/latex], and [latex]f(-4)[\/latex].\r\n\r\n[reveal-answer q=\"990936\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"990936\"]\r\n\r\nWe treat this like the previous problems, replacing [latex]x[\/latex] by the given inputs. For the first two, we get\r\n

      [latex]f(0)=\\frac{2(0)}{0+4}[\/latex]<\/p>\r\n

      [latex]f(0)=\\frac{0}{4}[\/latex]<\/p>\r\n

      [latex]f(0)=0[\/latex]<\/p>\r\nand\r\n

      [latex]f(3)=\\frac{2(3)}{3+4}[\/latex]<\/p>\r\n

      [latex]f(3)=\\frac{6}{7}[\/latex]<\/p>\r\nHowever, if we try plugging in [latex]x=-4[\/latex], we encounter an issue since division by 0 is undefined.\r\n

      [latex]f(-4)=\\frac{2(-4)}{-4+4}[\/latex]<\/p>\r\n

      [latex]f(-4)=\\frac{-8}{0}[\/latex]<\/p>\r\n

      [latex]f(-4)[\/latex] is undefined<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

      Finding the domain of a rational function<\/h2>\r\nThe previous example has an important implication about the domain of this fractional function, called a\u00a0rational function<\/strong>. Expanding upon our earlier definition of domain, the\u00a0domain of a function<\/strong> is the set of input values that lead to valid output values. It follows that for the function [latex]f(x)=\\frac{2x}{x+4}[\/latex], [latex]x=-4[\/latex] is not in the domain. Moreover, we can see that [latex]-4[\/latex] is the only value of [latex]x[\/latex] that results in a zero denominator, so the domain is all [latex]x[\/latex]-values except [latex]x=-4[\/latex]. In set-builder notation, we would write the domain as [latex]\\{x|x\\neq -4\\}[\/latex].\r\n\r\nWe conclude this section with an example that explicitly asks for the domain of a rational function.\r\n
      \r\n

      Example 10<\/h3>\r\nFind the domain of [latex]f(x)=\\displaystyle{\\frac{1}{3x+4}}[\/latex].\r\n\r\n[reveal-answer q=\"458615\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"458615\"]\r\n\r\nAs seen in the previous example, the problem we must avoid is a zero denominator. However, for this function, it may be less obvious what value of [latex]x[\/latex] results in a zero denominator. To find this, we set the denominator equal to zero and solve.\r\n

      [latex]3x+4=0[\/latex]<\/p>\r\n

      [latex]\\underline{\\hspace{.24in}-\\hspace{.02in} 4\\hspace{.04in}-4}[\/latex]<\/p>\r\n

      [latex]\\hspace{.4in} \\displaystyle{\\frac{3x}{3}=\\frac{-4}{3}}[\/latex]<\/p>\r\n

      [latex]\\hspace{.48in} x=-\\frac{4}{3}[\/latex]<\/p>\r\nRecall that we\u00a0don't<\/em> want the denominator to be zero, so this is the only value for [latex]x[\/latex] that is not in the domain. In set-builder notation, we would write this as\r\n

      [latex]D=\\left\\{x|x\\neq-\\frac{4}{3}\\right\\}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n ","rendered":"

      \n

      section 3.2 Learning Objectives<\/h3>\n

      3.2: Introduction to Functions<\/strong><\/p>\n

        \n
      • Determine if a relation is a function<\/li>\n
      • Identify the domain and range of a set of ordered pairs<\/li>\n
      • Evaluate a function written in function notation<\/li>\n
      • Find the domain of a rational function<\/li>\n<\/ul>\n<\/div>\n

         <\/p>\n

        There are many kinds of relations. A relation is simply a correspondence between sets of values or information. Think about members of your family and their ages. The pairing of each member of your family and their age is a relation. Each family member can be paired with an age in the set of ages of your family members. Another example of a relation is the pairing of a state with its U.S. senators. Each state can be matched with two individuals who have each been elected to serve as a senator. In turn, each senator can be matched with one specific state that he or she represents. Both of these are real-life examples of relations.<\/p>\n

        Determine if a relation is a function<\/h2>\n

        The first value of a relation is an input value and the second value is the output value. A function<\/strong> is a specific type of relation in which each input value has one and only one output value. An input is the independent<\/i> value, and the output is the dependent <\/i>value, as it depends on the value of the input.<\/p>\n

        Which set of values make up the input values and output values can affect whether or not the relation is a function. Notice in the first table below, where the input is \u201cname\u201d and the output is \u201cage,\u201d each input matches with exactly one output. This is an example of a function.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
        Family Member’s Name (Input)<\/th>\nFamily Member’s Age (Output)<\/th>\n<\/tr>\n<\/thead>\n
        Nellie<\/td>\n[latex]13[\/latex]<\/td>\n<\/tr>\n
        Marcos<\/td>\n[latex]11[\/latex]<\/td>\n<\/tr>\n
        Esther<\/td>\n[latex]46[\/latex]<\/td>\n<\/tr>\n
        Samuel<\/td>\n[latex]47[\/latex]<\/td>\n<\/tr>\n
        Nina<\/td>\n[latex]47[\/latex]<\/td>\n<\/tr>\n
        Paul<\/td>\n[latex]47[\/latex]<\/td>\n<\/tr>\n
        Katrina<\/td>\n[latex]21[\/latex]<\/td>\n<\/tr>\n
        Andrew<\/td>\n[latex]16[\/latex]<\/td>\n<\/tr>\n
        Maria<\/td>\n[latex]13[\/latex]<\/td>\n<\/tr>\n
        Ana<\/td>\n[latex]81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        Compare this with the next table where the input is \u201cage\u201d and the output is \u201cname.\u201d Some of the inputs result in more than one output. This is an example of a correspondence that is not <\/i>a function.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
        Family Member’s Age (Input)<\/th>\nFamily Member’s Name (Output)<\/th>\n<\/tr>\n<\/thead>\n
        [latex]11[\/latex]<\/td>\nMarcos<\/td>\n<\/tr>\n
        [latex]13[\/latex]<\/td>\nNellie, Maria<\/td>\n<\/tr>\n
        [latex]16[\/latex]<\/td>\nAndrew<\/td>\n<\/tr>\n
        [latex]21[\/latex]<\/td>\nKatrina<\/td>\n<\/tr>\n
        [latex]46[\/latex]<\/td>\nEsther<\/td>\n<\/tr>\n
        [latex]47[\/latex]<\/td>\nSamuel, Nina, Paul<\/td>\n<\/tr>\n
        [latex]81[\/latex]<\/td>\nAna<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        Now let us look at some other examples to determine whether the relations are functions or not and under what circumstances. Remember that a relation is a function if there is only one <\/strong><\/em>output for each input.<\/p>\n

        \n

        Example 1<\/h3>\n

        Fill in the table.<\/p>\n\n\n\n\n\n\n\n\n\n\n
        Input<\/th>\nOutput<\/th>\nFunction?<\/th>\nWhy or why not?<\/th>\n<\/tr>\n<\/thead>\n
        Name of senator<\/td>\nName of state<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
        Name of state<\/td>\nName of senator<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
        Time elapsed<\/td>\nHeight of a tossed ball<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
        Height of a tossed ball<\/td>\nTime elapsed<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
        Number of cars<\/td>\nNumber of tires<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
        Number of tires<\/td>\nNumber of cars<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
        Show Solution<\/span><\/p>\n
        \n\n\n\n\n\n\n\n\n\n\n
        Input<\/th>\nOutput<\/th>\nFunction?<\/th>\nWhy or why not?<\/th>\n<\/tr>\n<\/thead>\n
        Name of senator<\/td>\nName of state<\/td>\nYes<\/td>\nFor each input, there will only be one output because a senator only represents one state.<\/td>\n<\/tr>\n
        Name of state<\/td>\nName of senator<\/td>\nNo<\/td>\nFor each state that is an input, 2 names of senators would result because each state has two senators.<\/td>\n<\/tr>\n
        Time elapsed<\/td>\nHeight of a tossed ball<\/td>\nYes<\/td>\nAt a specific time, the ball has one specific height.<\/td>\n<\/tr>\n
        Height of a tossed ball<\/td>\nTime elapsed<\/td>\nNo<\/td>\nRemember that the ball was tossed up and fell down. So for a given height, there could be two different times when the ball was at that height. The input height can result in more than one output.<\/td>\n<\/tr>\n
        Number of cars<\/td>\nNumber of tires<\/td>\nYes<\/td>\nFor any input of a specific number of cars, there is one specific output representing the number of tires.<\/td>\n<\/tr>\n
        Number of tires<\/td>\nNumber of cars<\/td>\nYes<\/td>\nFor any input of a specific number of tires, there is one specific output representing the number of cars (assuming each car has all four of its tires).<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n

        Relations can be written as ordered pairs of numbers or as numbers in a table of values. Unless stated otherwise, the first number of an ordered pair is the x<\/em>-value and is the input, while the second number is the\u00a0y<\/em>-value and is the output.\u00a0\u00a0By examining the inputs (x<\/i>-coordinates) and outputs (y<\/i>-coordinates), you can determine whether or not the relation is a function. Remember, in a function, each input has only one output.<\/p>\n

        Identify the domain and range of a set of ordered pairs<\/h2>\n

        There is a name for the set of input values and another name for the set of output values for a function. The set of input values is called the domain of the function<\/b>. The set of output values is called the range of the function<\/b>.<\/p>\n

        If you have a set of ordered pairs, you can find the domain by listing all of the input values, which are the x<\/i>-coordinates. To find the range, list all of the output values, which are the y<\/i>-coordinates.<\/p>\n

        Consider the following set of ordered pairs:<\/p>\n

        [latex]\\{(\u22122,0),(0,6),(2,12),(4,18)\\}[\/latex]<\/p>\n

        You have the following:<\/p>\n

        [latex]\\begin{array}{l}\\text{Domain:}\\{\u22122,0,2,4\\}\\\\\\text{Range:}\\{0,6,12,18\\}\\end{array}[\/latex]<\/p>\n

        Now try it yourself.<\/p>\n

        \n

        Example 2<\/h3>\n

        List the domain and range for the following table of values where x<\/em> is the input and y<\/em> is the output. Then determine if the relation is a function.<\/p>\n\n\n\n\n\n\n\n\n
        x<\/i><\/th>\ny<\/i><\/th>\n<\/tr>\n
        [latex]\u22123[\/latex]<\/td>\n[latex]4[\/latex]<\/td>\n<\/tr>\n
        [latex]\u22122[\/latex]<\/td>\n[latex]4[\/latex]<\/td>\n<\/tr>\n
        [latex]\u22121[\/latex]<\/td>\n[latex]4[\/latex]<\/td>\n<\/tr>\n
        [latex]2[\/latex]<\/td>\n[latex]4[\/latex]<\/td>\n<\/tr>\n
        [latex]3[\/latex]<\/td>\n[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
        Show Solution<\/span><\/p>\n
        \n

        The domain describes all the inputs, and we can use set notation with brackets { } to make the list.<\/p>\n

        [latex]\\text{Domain}:\\{-3,-2,-1,2,3\\}[\/latex]<\/p>\n

        The range describes all the outputs.<\/p>\n

        [latex]\\text{Range}:\\{4\\}[\/latex]<\/p>\n

        We only listed\u00a0[latex]4[\/latex] once because it is not necessary to list a number every time it appears in a set.<\/p>\n

        Each input has only one output, and the fact that it is the same output (4) does not matter.<\/p>\n

        Therefore, this relation is a function.<\/p>\n

        A helpful way to visualize this is through an “arrow diagram.”\u00a0 We list our domain and range (and as mentioned above with our range, do not list an element more than once in either set). Then, for each input in the range, we draw an arrow to the output it maps to in the range. This table of values corresponds to the arrow diagram shown below.<\/p>\n

        \"Mapping<\/p>\n

        Since each input maps to only one output (equivalently, there is only one arrow coming from each input), the relation is a function.<\/p>\n

        Answer<\/span><\/h4>\n

        [latex]\\text{Domain}:\\{-3,-2,-1,2,3\\}[\/latex]<\/p>\n

        [latex]\\text{Range}:\\{4\\}[\/latex]<\/p>\n

        The relation is a function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

        In the following video we provide another example of identifying whether a table of values represents a function as well as determining the domain and range of each.<\/p>\n