{"id":16606,"date":"2019-07-30T15:38:15","date_gmt":"2019-07-30T15:38:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/inverse-functions-2\/"},"modified":"2022-01-09T19:12:50","modified_gmt":"2022-01-09T19:12:50","slug":"inverse-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/inverse-functions\/","title":{"raw":"Inverse Functions","rendered":"Inverse Functions"},"content":{"raw":"

A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.<\/p>\r\n

If some physical machines can run in two directions, we might ask whether some of the function \u201cmachines\u201d we have been studying can also run backwards. Figure below provides a visual representation of this question. In this section, we will consider the reverse nature of functions.<\/p>\r\n\r\n

\r\n
Can a function \u201cmachine\u201d operate in reverse?<\/div>\r\n\"Diagram<\/span>\r\n\r\n<\/div>\r\n
\r\n

Verifying That Two Functions Are Inverse Functions<\/h3>\r\n

Suppose a fashion designer traveling to Milan for a fashion show wants to know what the temperature will be. He is not familiar with the Celsius<\/span> scale. To get an idea of how temperature measurements are related, he asks his assistant, Betty, to convert 75 degrees Fahrenheit<\/span> to degrees Celsius. She finds the formula<\/p>\r\n\r\n

[latex]C=\\dfrac{5}{9}\\left(F-32 \\right)[\/latex]<\/div>\r\n

and substitutes 75 for F to calculate<\/p>\r\n\r\n

[latex]\\dfrac{5}{9}\\left(75-32\\right)\\approx 24 \u00b0C .[\/latex]<\/div>\r\n

Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, he sends his assistant the week\u2019s weather forecast for Milan, and asks her to convert all of the temperatures to degrees Fahrenheit.<\/p>\r\n\r\n

\"A<\/span><\/div>\r\nAt first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for F after substituting a value for C. For example, to convert 26 degrees Celsius, she could write\r\n
[latex] \\begin{array}{ccc}\\hfill 26& =& \\frac{5}{9}\\left(F-32\\right)\\hfill \\\\ \\hfill 26\\cdot \\frac{9}{5}& =& F-32\\hfill \\\\ \\hfill F& =& 26\\cdot \\frac{9}{5}+32\\approx 79\\hfill \\end{array} [\/latex]<\/div>\r\n

After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.<\/p>\r\n

The formula for which Betty is searching corresponds to the idea of an inverse function<\/strong>, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.<\/p>\r\nInformally, this means that inverse functions \u201cundo\u201d each other. However, just as zero does not have a reciprocal<\/span>, some functions do not have inverses.\r\n

Given a function [latex] f(x) [\/latex] we represent its inverse as [latex]\u00a0f^{-1} (x)<\/span>\u00a0[\/latex], read as \u201cf inverse of x\"<\/em>. The raised -1<\/em> is part of the notation. It is not an exponent; it does not imply a power of -1. In other words, [latex]\u00a0f^{-1} (x)<\/span>\u00a0[\/latex] does <\/span>not<\/em> mean\u00a0[latex] \\frac{1}{f(x)}\u00a0[\/latex] because\u00a0[latex] \\frac{1}{f(x)}\u00a0[\/latex] is the reciprocal of\u00a0[latex] f(x)[\/latex] and not the inverse.<\/span><\/p>\r\n[embed]https:\/\/www.youtube.com\/watch?v=Hbmx7TPCar4[\/embed]\r\n

\r\n
\r\n
\r\n

Inverse Function<\/h2>\r\n

For any one-to-one function [latex]f(x)=y[\/latex]\u00a0<\/span>a function [latex]\u00a0f^{-1} (x)<\/span>\u00a0[\/latex]\u00a0<\/span>is an inverse function<\/span> of [latex]\u00a0f<\/span>\u00a0[\/latex]<\/span> if [latex]\u00a0f^{-1} (y)=x<\/span>\u00a0[\/latex].\u00a0<\/span>This can also be written as [latex]\u00a0f^{-1} (f(x))=x<\/span>\u00a0[\/latex]\u00a0<\/span>for all\u00a0 [latex] x [\/latex] in the domain of [latex] f [\/latex]. It also follows that [latex] f(f^{-1} (x))=x<\/span>\u00a0[\/latex]\u00a0<\/span>for all [latex] x [\/latex] in the domain of [latex] f^{-1} [\/latex] if\u00a0[latex]\u00a0f^{-1} (x)<\/span>\u00a0[\/latex]<\/span>\u00a0is the inverse of\u00a0[latex]\u00a0f<\/span>\u00a0[\/latex].<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
\r\n

Identifying an Inverse Function for a Given Input-Output Pair<\/h2>\r\n

If for a particular one-to-one function\u00a0[latex]f(2)=4[\/latex]\u00a0<\/span>and\u00a0[latex]f(5)=12[\/latex]\u00a0<\/span>what are the corresponding input and output values for the inverse function?<\/p>\r\n\r\n<\/div>\r\n

\r\n

The inverse function reverses the input and output quantities, so if<\/p>\r\n\r\n

[latex]\\begin{array}{ccc}\\hfill f(2)& =& 4,\\text{\u00a0then\u00a0} f^{-1}(4)=2;\\hfill \\\\ \\hfill f(5)& =& 12, text{\u00a0then } f^{-1}(12)=5.\\hfill \\end{array}[\/latex]<\/div>\r\n

Alternatively, if we want to name the inverse function\u00a0[latex]g[\/latex]\u00a0<\/span>then\u00a0[latex]g(4)=2[\/latex]\u00a0<\/span>and\u00a0[latex]g(12)=5[\/latex] .<\/span><\/p>\r\n\r\n<\/div>\r\n

\r\n

Analysis<\/h4>\r\n

Notice that if we show the coordinate pairs in a table form, the input and output are clearly reversed.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex](x,f(x))[\/latex]<\/th>\r\n[latex](x,g(x))[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
(2,4)<\/td>\r\n(4,2)<\/td>\r\n<\/tr>\r\n
(5,12)<\/td>\r\n(12,5)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n

Is it possible for a function to have more than one inverse?<\/strong><\/p>\r\n

No. If two supposedly different functions, say g and h, both meet the definition of being inverses of another function f, then you can prove that g=h. On any one domain, the original function has only one unique inverse.<\/em><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n

Finding and Evaluating Inverse Functions<\/h3>\r\n

Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.<\/p>\r\n\r\n

\r\n

Inverting Tabular Functions<\/h4>\r\n

Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.<\/p>\r\n

Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.<\/p>\r\n\r\n

\r\n

Example:\u00a0Interpreting the Inverse of a Tabular Function<\/h3>\r\n

A function\u00a0[latex]f(t)[\/latex]\u00a0<\/span>is given in Figure below, showing distance in miles that a car has traveled in t<\/em> minutes. Find and interpret\u00a0[latex]f^{-1}[\/latex] .<\/span><\/p>\r\n\r\n<\/colgroup>\r\n\r\n\r\n\r\n
t, minutes<\/strong><\/td>\r\n30<\/td>\r\n50<\/td>\r\n70<\/td>\r\n90<\/td>\r\n<\/tr>\r\n
f(t), miles<\/strong><\/td>\r\n20<\/td>\r\n40<\/td>\r\n60<\/td>\r\n70<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"574523\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"574523\"]The inverse function takes an output of f and returns an input for f. So in the expression\u00a0[latex]f^{-1}(70)[\/latex], 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function f, 90 minutes, so\u00a0[latex]f^{-1}(70)=90[\/latex].\u00a0The interpretation of this is that, to drive 70 miles, it took 90 minutes.[\/hidden-answer]\r\n\r\n<\/div>\r\n
\r\n

Try it<\/h3>\r\n
\r\n

Using previous Figure, find and interpret (a)\u00a0[latex]f(60)[\/latex],\u00a0<\/span>and (b)\u00a0[latex]f^{-1}(60)[\/latex].<\/span><\/p>\r\n\r\n[reveal-answer q=\"18178\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"18178\"]\r\n\r\na. [latex]f(60)=50[\/latex]\u00a0In 60 minutes, 50 miles are traveled.\r\n\r\nb. [latex]f^{-1}(60)=70[\/latex]\u00a0To travel 60 miles, it will take 70 minutes.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n
\r\n\r\nGiven the graph of a function, evaluate its inverse at specific points.<\/strong>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n
    \r\n \t
  1. Find the desired input on the y<\/em>-axis of the given graph.<\/li>\r\n \t
  2. Read the inverse function\u2019s output from the x<\/em>-axis of the given graph.<\/li>\r\n<\/ol>\r\n
    \r\n

    ExAMPLE:\u00a0Evaluating a Function and Its Inverse from a Graph at Specific Points<\/h3>\r\n

    A function\u00a0[latex]g(x)[\/latex]\u00a0<\/span>is given below. Find\u00a0[latex]g(3)[\/latex],\u00a0<\/span>and\u00a0[latex]g^{-1}(3)[\/latex].<\/span><\/p>\r\n\r\n

    \"Graph<\/span><\/div>\r\n
    \r\n[reveal-answer q=\"362132\"]Show Answer[\/reveal-answer]<\/div>\r\n
    [hidden-answer a=\"362132\"]<\/div>\r\n
    \r\n

    To evaluate\u00a0[latex]g(3)[\/latex]\u00a0<\/span>we find 3 on the x<\/em>-axis and find the corresponding output value on the y<\/em>-axis. The point (3,1) tells us that\u00a0[latex]g(3)=1[\/latex].<\/span><\/p>\r\n

    To evaluate\u00a0[latex]g^{-1}(3)[\/latex]\u00a0<\/span>recall that by definition\u00a0[latex]g^{-1}(3)[\/latex]\u00a0<\/span>means the value of x<\/em> for which\u00a0[latex]g(x)=3[\/latex].\u00a0<\/span>By looking for the output value 3 on the vertical axis, we find the point (5,3) on the graph, which means\u00a0[latex]g(5)=3[\/latex]\u00a0<\/span>so by definition,\u00a0[latex]g^{-1}(3)=5[\/latex].\u00a0<\/span>See Figure below.<\/p>\r\n\r\n

    \"Graph<\/span><\/div>\r\n<\/div>\r\n
    \r\n[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
    \r\n\r\n \r\n

    Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula, for example y as a function of x, we can often find the inverse function by solving the formula to obtain x as a function of y.<\/p>\r\n\r\n

    \r\n

    Given a function represented by a formula, find the inverse.<\/strong><\/p>\r\n\r\n

      \r\n \t
    1. Make sure\u00a0[latex]f(x)=y[\/latex]\u00a0<\/span> is a one-to-one function.<\/li>\r\n \t
    2. Solve for x<\/em>.<\/li>\r\n \t
    3. Interchange x<\/em> and y<\/em>.<\/li>\r\n<\/ol>\r\n
      \r\n

      Example:\u00a0Inverting the Fahrenheit-to-Celsius Function<\/h3>\r\n

      Given\u00a0[latex]<\/span>C=\\frac{5}{9}\\left(F-32\\right)<\/span>[\/latex], f<\/span>ind a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.<\/p>\r\n\r\n

      \r\n[reveal-answer q=\"135755\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"135755\"]<\/div>\r\n
      \r\n
      \\begin{array}{ccc}\\hfill C& =& \\frac{5}{9}\\left(F-32\\right)\\hfill \\\\<\/div>\r\n
      \\hfill C\\cdot \\frac{9}{5}& =& F-32\\hfill \\\\<\/div>\r\n
      \\hfill F& =& \\frac{9}{5}C+32\\hfill \\end{array})<\/div>\r\n

      By solving in general, we have uncovered the inverse function. If\u00a0[latex]<\/span>Ch(F)=\\frac{5}{9}\\left(F-32\\right)<\/span>[\/latex]<\/span><\/p>\r\n

      then<\/p>\r\n\r\n

      [latex]F={h}^{-1}\\left(C\\right)=\\frac{9}{5}C+32[\/latex].<\/div>\r\n

      In this case, we introduced a function h to represent the conversion because the input and output variables are descriptive, and writing [latex]{C}^{-1}[\/latex] could get confusing.<\/p>\r\n\r\n<\/div>\r\n

      [\/hidden-answer]<\/div>\r\n<\/div>\r\n
      \r\n
      \r\n

      try it<\/h3>\r\nSolve for x in terms of y, given [latex]y=\\frac{1}{3}\\left(x-5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"480227\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"480227\"][latex]x=3y+5[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video for a review of the concept of inverse functions.\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=q6y0ToEhT1E[\/embed]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
      \r\n

      Key Concepts<\/h3>\r\n
        \r\n \t
      • For a function to have an inverse, it must be one-to-one (pass the horizontal line test).<\/li>\r\n \t
      • For a tabular function, exchange the input and output rows to obtain the inverse.<\/li>\r\n \t
      • The inverse of a function can be determined at specific points on its graph.<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"

        A reversible heat pump is a climate-control system that is an air conditioner and a heater in a single device. Operated in one direction, it pumps heat out of a house to provide cooling. Operating in reverse, it pumps heat into the building from the outside, even in cool weather, to provide heating. As a heater, a heat pump is several times more efficient than conventional electrical resistance heating.<\/p>\n

        If some physical machines can run in two directions, we might ask whether some of the function \u201cmachines\u201d we have been studying can also run backwards. Figure below provides a visual representation of this question. In this section, we will consider the reverse nature of functions.<\/p>\n

        \n
        Can a function \u201cmachine\u201d operate in reverse?<\/div>\n

        \"Diagram<\/span><\/p>\n<\/div>\n

        \n

        Verifying That Two Functions Are Inverse Functions<\/h3>\n

        Suppose a fashion designer traveling to Milan for a fashion show wants to know what the temperature will be. He is not familiar with the Celsius<\/span> scale. To get an idea of how temperature measurements are related, he asks his assistant, Betty, to convert 75 degrees Fahrenheit<\/span> to degrees Celsius. She finds the formula<\/p>\n

        [latex]C=\\dfrac{5}{9}\\left(F-32 \\right)[\/latex]<\/div>\n

        and substitutes 75 for F to calculate<\/p>\n

        [latex]\\dfrac{5}{9}\\left(75-32\\right)\\approx 24 \u00b0C .[\/latex]<\/div>\n

        Knowing that a comfortable 75 degrees Fahrenheit is about 24 degrees Celsius, he sends his assistant the week\u2019s weather forecast for Milan, and asks her to convert all of the temperatures to degrees Fahrenheit.<\/p>\n

        \"A<\/span><\/div>\n

        At first, Betty considers using the formula she has already found to complete the conversions. After all, she knows her algebra, and can easily solve the equation for F after substituting a value for C. For example, to convert 26 degrees Celsius, she could write<\/p>\n

        [latex]\\begin{array}{ccc}\\hfill 26& =& \\frac{5}{9}\\left(F-32\\right)\\hfill \\\\ \\hfill 26\\cdot \\frac{9}{5}& =& F-32\\hfill \\\\ \\hfill F& =& 26\\cdot \\frac{9}{5}+32\\approx 79\\hfill \\end{array}[\/latex]<\/div>\n

        After considering this option for a moment, however, she realizes that solving the equation for each of the temperatures will be awfully tedious. She realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one that takes the Celsius temperature and outputs the Fahrenheit temperature.<\/p>\n

        The formula for which Betty is searching corresponds to the idea of an inverse function<\/strong>, which is a function for which the input of the original function becomes the output of the inverse function and the output of the original function becomes the input of the inverse function.<\/p>\n

        Informally, this means that inverse functions \u201cundo\u201d each other. However, just as zero does not have a reciprocal<\/span>, some functions do not have inverses.<\/p>\n

        Given a function [latex]f(x)[\/latex] we represent its inverse as [latex]\u00a0f^{-1} (x)<\/span>\u00a0[\/latex], read as \u201cf inverse of x”<\/em>. The raised -1<\/em> is part of the notation. It is not an exponent; it does not imply a power of -1. In other words, [latex]\u00a0f^{-1} (x)<\/span>\u00a0[\/latex] does <\/span>not<\/em> mean\u00a0[latex]\\frac{1}{f(x)}\u00a0[\/latex] because\u00a0[latex]\\frac{1}{f(x)}\u00a0[\/latex] is the reciprocal of\u00a0[latex]f(x)[\/latex] and not the inverse.<\/span><\/p>\n