Inverse Functions

Overview

An inverse function is a function that "reverses" the effect of the original function. If a function \( f(x) \) takes an input \( x \) and gives an output \( y \), then its inverse function \( f^{-1}(x) \) takes the output \( y \) and returns the original input \( x \). In simple terms, an inverse function undoes the operation of the original function.

Notation and Definition

The inverse of a function \( f(x) \) is denoted as \( f^{-1}(x) \). For the functions \( f \) and \( f^{-1} \), we have the following property:

\( f(f^{-1}(x)) = x \)

This means that applying the inverse function \( f^{-1} \) to the result of \( f(x) \) will give back the original input \( x \).

Conditions for Inverses

Not all functions have an inverse. For a function to have an inverse, it must be one-to-one, meaning that each output corresponds to exactly one input. A function that is one-to-one passes the horizontal line test: if any horizontal line intersects the graph of the function at more than one point, the function does not have an inverse.

Finding the Inverse of a Function

To find the inverse of a function, follow these steps:

  1. Write the function as \( y = f(x) \).
  2. Swap \( x \) and \( y \) in the equation.
  3. Solve for \( y \) in terms of \( x \). The resulting equation represents \( f^{-1}(x) \).

Example

Let \( f(x) = 2x + 3 \). To find the inverse:

  1. Write \( y = 2x + 3 \).
  2. Swap \( x \) and \( y \): \( x = 2y + 3 \).
  3. Solve for \( y \):

    4. \( x - 3 = 2y \rightarrow y = \frac{x - 3}{2} \)

    So, the inverse function is \( f^{-1}(x) = \frac{x - 3}{2} \).

Inverse Functions and Graphs

The graph of an inverse function is a reflection of the graph of the original function across the line \( y = x \). If you take any point \( (a, b) \) on the graph of \( f(x) \), then the point \( (b, a) \) will be on the graph of \( f^{-1}(x) \).

Practice Questions

  1. Find the inverse of the function \( f(x) = 3x - 5 \).
    Solution

    First, write \( y = 3x - 5 \). Swap \( x \) and \( y \):

    \( x = 3y - 5 \)

    Solve for \( y \):

    \( x + 5 = 3y \rightarrow y = \frac{x + 5}{3} \)

    So, the inverse function is \( f^{-1}(x) = \frac{x + 5}{3} \).

  2. Find the inverse of the function \( f(x) = \frac{x + 2}{4} \).
    Solution

    First, write \( y = \frac{x + 2}{4} \). Swap \( x \) and \( y \):

    \( x = \frac{y + 2}{4} \)

    Solve for \( y \):

    \( 4x = y + 2 \rightarrow y = 4x - 2 \)

    So, the inverse function is \( f^{-1}(x) = 4x - 2 \).

  3. Find the inverse of the function \( f(x) = \sqrt{x + 1} \).
    Solution

    First, write \( y = \sqrt{x + 1} \). Swap \( x \) and \( y \):

    \( x = \sqrt{y + 1} \)

    Solve for \( y \):

    \( x^2 = y + 1 \rightarrow y = x^2 - 1 \)

    So, the inverse function is \( f^{-1}(x) = x^2 - 1 \).

  4. Find the inverse of the function \( f(x) = \frac{1}{x - 3} \).
    Solution

    First, write \( y = \frac{1}{x - 3} \). Swap \( x \) and \( y \):

    \( x = \frac{1}{y - 3} \)

    Solve for \( y \):

    \( x(y - 3) = 1 \rightarrow xy - 3x = 1 \rightarrow xy = 3x + 1 \rightarrow y = \frac{3x + 1}{x} \)

    So, the inverse function is \( f^{-1}(x) = \frac{3x + 1}{x} \).

  5. Verify whether the functions \( f(x) = 2x + 1 \) and \( g(x) = \frac{x - 1}{2} \) are inverses of each other.
    Solution

    To verify that two functions are inverses, check if \( f(g(x)) = x \) and \( g(f(x)) = x \).

    \( f(g(x)) = f\left(\frac{x - 1}{2}\right) = 2\left(\frac{x - 1}{2}\right) + 1 = x - 1 + 1 = x \)

    \( g(f(x)) = g(2x + 1) = \frac{(2x + 1) - 1}{2} = \frac{2x}{2} = x \)

    Since both conditions are satisfied, \( f(x) \) and \( g(x) \) are inverses of each other.

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