{"id":720,"date":"2025-07-14T21:31:49","date_gmt":"2025-07-14T21:31:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=720"},"modified":"2026-01-08T22:44:57","modified_gmt":"2026-01-08T22:44:57","slug":"720","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/720\/","title":{"raw":"Inverse Functions: Learn It 3","rendered":"Inverse Functions: Learn It 3"},"content":{"raw":"

Finding and Evaluating Inverse Functions<\/h2>\r\nOnce 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.\r\n

Inverting Tabular Functions<\/h3>\r\nSuppose 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.\r\n\r\nEach 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.\r\n\r\n
A function [latex]f\\left(t\\right)[\/latex] is given\u00a0below, showing distance in miles that a car has traveled in [latex]t[\/latex] minutes. Find and interpret [latex]{f}^{-1}\\left(70\\right)[\/latex].\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]t[\/latex] (minutes)<\/th>\r\n[latex]f(t)[\/latex] (miles)<\/th>\r\n<\/tr>\r\n
[latex]30[\/latex]<\/td>\r\n[latex]20[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]50[\/latex]<\/td>\r\n[latex]40[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]70[\/latex]<\/td>\r\n[latex]60[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]90[\/latex]<\/td>\r\n[latex]70[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"61261\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"61261\"]\r\n\r\nThe inverse function takes an output of [latex]f[\/latex] and returns an input for [latex]f[\/latex]. So in the expression [latex]{f}^{-1}\\left(70\\right)[\/latex], [latex]70[\/latex] is an output value of the original function, representing [latex]70[\/latex] miles. The inverse will return the corresponding input of the original function [latex]f[\/latex], [latex]90[\/latex] minutes, so [latex]{f}^{-1}\\left(70\\right)=90[\/latex]. The interpretation of this is that, to drive [latex]70[\/latex] miles, it took [latex]90[\/latex] minutes.\r\n\r\nAlternatively, recall that the definition of the inverse was that if [latex]f\\left(a\\right)=b[\/latex], then [latex]{f}^{-1}\\left(b\\right)=a[\/latex]. By this definition, if we are given [latex]{f}^{-1}\\left(70\\right)=a[\/latex], then we are looking for a value [latex]a[\/latex] so that [latex]f\\left(a\\right)=70[\/latex]. In this case, we are looking for a [latex]t[\/latex] so that [latex]f\\left(t\\right)=70[\/latex], which is when [latex]t=90[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
[ohm_question hide_question_numbers=1]317945[\/ohm_question]<\/section>\r\n

Evaluating the Inverse of a Function, Given a Graph of the Original Function<\/h3>\r\nThe domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical<\/em> extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal<\/em> extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function\u2019s graph.\r\n\r\n
How To: Given the graph of a function, evaluate its inverse at specific points.<\/strong>\r\n
    \r\n \t
  1. Find the desired input of the inverse function on the [latex]y[\/latex]-axis of the given graph.<\/li>\r\n \t
  2. Read the inverse function\u2019s output from the [latex]x[\/latex]-axis of the given graph.<\/li>\r\n<\/ol>\r\n<\/section>
    A function [latex]g\\left(x\\right)[\/latex] is given\u00a0below. Find [latex]g\\left(3\\right)[\/latex] and [latex]{g}^{-1}\\left(3\\right)[\/latex].\"Graph[reveal-answer q=\"334632\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"334632\"]To evaluate [latex]g\\left(3\\right)[\/latex], we find [latex]3[\/latex] on the [latex]x[\/latex]-axis and find the corresponding output value on the [latex]y[\/latex]-axis. The point [latex]\\left(3,1\\right)[\/latex] tells us that [latex]g\\left(3\\right)=1[\/latex].\r\n[latex]\\\\[\/latex]\r\nTo evaluate [latex]{g}^{-1}\\left(3\\right)[\/latex], recall that by definition [latex]{g}^{-1}\\left(3\\right)[\/latex] means the value of [latex]x[\/latex] for which [latex]g\\left(x\\right)=3[\/latex]. By looking for the output value [latex]3[\/latex] on the vertical axis, we find the point [latex]\\left(5,3\\right)[\/latex] on the graph, which means [latex]g\\left(5\\right)=3[\/latex], so by definition, [latex]{g}^{-1}\\left(3\\right)=5[\/latex].\"Graph[\/hidden-answer]<\/section>
    [ohm_question hide_question_numbers=1]317947[\/ohm_question]<\/section>\r\n

    Finding Inverses of Functions Represented by Formulas<\/h3>\r\nSometimes 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\u2014for example, [latex]y[\/latex] as a function of [latex]x-[\/latex] we can often find the inverse function by solving to obtain [latex]x[\/latex] as a function of [latex]y[\/latex].\r\n\r\n
    How To: Given a function represented by a formula, find the inverse.<\/strong>\r\n
      \r\n \t
    1. Verify that\u00a0[latex]f[\/latex] is a one-to-one function.<\/li>\r\n \t
    2. Replace [latex]f\\left(x\\right)[\/latex] with [latex]y[\/latex].<\/li>\r\n \t
    3. Interchange [latex]x[\/latex]\u00a0and [latex]y[\/latex].<\/li>\r\n \t
    4. Solve for [latex]y[\/latex], and rename the function [latex]{f}^{-1}\\left(x\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>
      Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.\r\n

      [latex]C=\\frac{5}{9}\\left(F - 32\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"625400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"625400\"]\r\n

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

      [latex]C=h\\left(F\\right)=\\frac{5}{9}\\left(F - 32\\right)[\/latex],<\/p>\r\nthen\r\n

      [latex]F={h}^{-1}\\left(C\\right)=\\frac{9}{5}C+32[\/latex].<\/p>\r\nIn this case, we introduced a function [latex]h[\/latex] to represent the conversion because the input and output variables are descriptive, and writing [latex]{C}^{-1}[\/latex] could get confusing.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

      Find the inverse of the function [latex]f\\left(x\\right)=\\dfrac{2}{x - 3}+4[\/latex].\r\n

      [reveal-answer q=\"903464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"903464\"]<\/p>\r\n

      [latex]\\begin{align}&y=\\frac{2}{x - 3}+4 && \\text{Change }f(x)\\text{ to }y. \\\\[1.5mm]&x=\\frac{2}{y - 3}+4 && \\text{Switch }x\\text{ and }y. \\\\[1.5mm] &y - 4=\\frac{2}{x - 3} && \\text{Subtract 4 from both sides}. \\\\[1.5mm] &y - 3=\\frac{2}{x - 4} && \\text{Multiply both sides by }y - 3\\text{ and divide by }x - 4. \\\\[1.5mm] &y=\\frac{2}{x - 4}+3 && \\text{Add 3 to both sides}.\\\\[-3mm]&\\end{align}[\/latex]<\/p>\r\n

      So [latex]{f}^{-1}\\left(x\\right)=\\dfrac{2}{x - 4}+3[\/latex].<\/p>\r\nAnalysis of the Solution<\/strong>\r\n\r\nThe domain and range of [latex]f[\/latex] exclude the values [latex]3[\/latex] and [latex]4[\/latex], respectively. [latex]f[\/latex] and [latex]{f}^{-1}[\/latex] are equal at two points but are not the same function, as we can see by creating\u00a0the table below.\r\n\r\n\r\n\r\n\r\n
      [latex]x[\/latex]<\/strong><\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n[latex]{f}^{-1}\\left(y\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
      [latex]f\\left(x\\right)[\/latex] <\/strong><\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n[latex]y[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/section>
      [ohm_question hide_question_numbers=1]317948[\/ohm_question]<\/section>","rendered":"

      Finding and Evaluating Inverse Functions<\/h2>\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>\n

      Inverting Tabular Functions<\/h3>\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>\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>\n

      A function [latex]f\\left(t\\right)[\/latex] is given\u00a0below, showing distance in miles that a car has traveled in [latex]t[\/latex] minutes. Find and interpret [latex]{f}^{-1}\\left(70\\right)[\/latex].<\/p>\n\n\n\n\n\n\n\n
      [latex]t[\/latex] (minutes)<\/th>\n[latex]f(t)[\/latex] (miles)<\/th>\n<\/tr>\n
      [latex]30[\/latex]<\/td>\n[latex]20[\/latex]<\/td>\n<\/tr>\n
      [latex]50[\/latex]<\/td>\n[latex]40[\/latex]<\/td>\n<\/tr>\n
      [latex]70[\/latex]<\/td>\n[latex]60[\/latex]<\/td>\n<\/tr>\n
      [latex]90[\/latex]<\/td>\n[latex]70[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n