{"id":1137,"date":"2024-04-11T16:52:46","date_gmt":"2024-04-11T16:52:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&p=1137"},"modified":"2024-08-05T12:27:45","modified_gmt":"2024-08-05T12:27:45","slug":"inverse-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/inverse-functions-learn-it-2\/","title":{"raw":"Inverse Functions: Learn It 2","rendered":"Inverse Functions: Learn It 2"},"content":{"raw":"

Finding a Function\u2019s Inverse<\/h2>\r\n

To find the inverse of a function, you first ensure the function is one-to-one.\u00a0<\/p>\r\n

When given a one-to-one function, to find its inverse, solve the equation[latex] y=f(x)[\/latex] for [latex]x[\/latex], and then swap the roles of [latex]x[\/latex] and [latex]y[\/latex]. The new equation [latex]x=f^{\u22121}(y)[\/latex] represents the inverse function [latex]f^{-1}[\/latex], which switches the original function\u2019s inputs and outputs. This process is essential when plotting both the function and its inverse on the same graph, as their coordinates are reflections of each other across the line [latex]y=x[\/latex].<\/p>\r\n

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How to: Find an Inverse Function<\/strong><\/p>\r\n

    \r\n\t
  1. Solve the equation [latex]y=f(x)[\/latex] for [latex]x[\/latex].<\/li>\r\n\t
  2. Interchange the variables [latex]x[\/latex] and [latex]y[\/latex] and write [latex]y=f^{-1}(x)[\/latex].<\/li>\r\n<\/ol>\r\n<\/section>\r\n

    To complete the first step to finding an inverse function, we must isolate a variable in a given equation.<\/p>\r\n

    \r\n

    Recall Isolating a Variable in a Formula<\/strong><\/p>\r\n

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    1. Identify the variable you want to isolate and the terms it's associated with.<\/li>\r\n\t
    2. Use inverse operations to 'undo' any arithmetic or algebraic actions applied to the variable (addition is undone by subtraction, multiplication by division, etc.).<\/li>\r\n\t
    3. Perform the same operation on both sides of the equation to maintain equality.<\/li>\r\n\t
    4. Repeat the process until the variable is by itself on one side of the equation.<\/li>\r\n\t
    5. Simplify the equation as needed to achieve the simplest form with the variable isolated.<\/li>\r\n<\/ol>\r\n<\/section>\r\n
      \r\n

      Find the inverse for the function [latex]f(x)=3x-4[\/latex]. State the domain and range of the inverse function. Verify that [latex]f^{-1}(f(x))=x[\/latex].<\/p>\r\n

      [reveal-answer q=\"fs-id1170572481500\"]Show Solution[\/reveal-answer]
      \r\n[hidden-answer a=\"fs-id1170572481500\"]<\/p>\r\n

      Follow the steps outlined in the strategy.<\/p>\r\n

      Step 1. If [latex]y=3x-4[\/latex], then [latex]3x=y+4[\/latex] and [latex]x=\\frac{1}{3}y+\\frac{4}{3}[\/latex].<\/p>\r\n

      Step 2. Rewrite as [latex]y=\\frac{1}{3}x+\\frac{4}{3}[\/latex] and let [latex]y=f^{-1}(x)[\/latex].<\/p>\r\n

      Therefore, [latex]f^{-1}(x)=\\frac{1}{3}x+\\frac{4}{3}[\/latex].<\/p>\r\n

      Since the domain of [latex]f[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex], the range of [latex]f^{-1}[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex]. Since the range of [latex]f[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex], the domain of [latex]f^{-1}[\/latex] is [latex](\u2212\\infty ,\\infty)[\/latex].<\/p>\r\n

      You can verify that [latex]f^{-1}(f(x))=x[\/latex] by writing<\/p>\r\n

      [latex]f^{-1}(f(x))=f^{-1}(3x-4)=\\frac{1}{3}(3x-4)+\\frac{4}{3}=x-\\frac{4}{3}+\\frac{4}{3}=x[\/latex].<\/div>\r\n

       <\/p>\r\n

      Note that for [latex]f^{-1}(x)[\/latex] to be the inverse of [latex]f(x)[\/latex], both [latex]f^{-1}(f(x))=x[\/latex] and [latex]f(f^{-1}(x))=x[\/latex] for all [latex]x[\/latex] in the domain of the inside function.<\/p>\r\n\r\nWatch the following video to see the worked solution to this example.