Inverse Functions: Learn It 1

  • Confirm that two functions are inverses
  • Figure out the allowed inputs and outputs for an inverse function and adjust the original function’s domain to ensure it is one-to-one
  • Discover or calculate the inverse of a function
  • Draw the inverse of a function on the same graph by reflecting it across the line [latex]y=x[/latex]

Inverse Function

Betty is traveling to Milan for a fashion show and wants to know what the temperature will be. She is not familiar with the Celsius scale. To get an idea of how temperature measurements are related, Betty wants to convert 75 degrees Fahrenheit to degrees Celsius using the formula

[latex]C = \frac{5}{9}(F - 32)[/latex]

and substitutes [latex]75[/latex] for [latex]F[/latex] to calculate

[latex]\frac{5}{9}(75 - 32) \approx 24^\circ C[/latex]

Knowing that a comfortable [latex]75[/latex] degrees Fahrenheit is about [latex]24[/latex] degrees Celsius, Betty gets the week’s weather forecast for Milan, and wants to convert all of the temperatures to degrees Fahrenheit.

A forecast of Monday’s through Thursday’s weather.

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 [latex]F[/latex] after substituting a value for [latex]C[/latex]. For example, to convert [latex]26[/latex] degrees Celsius, she could write:

[latex]\begin{array}{rcl} 26 & = & \frac{5}{9}(F - 32) \\ 26 \cdot \frac{9}{5} & = & F - 32 \\ F & = & 26 \cdot \frac{9}{5} + 32 \approx 79 \end{array}[/latex]

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.

The formula for which Betty is searching corresponds to the idea of an inverse function, 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.

Given a function [latex]f(x)[/latex], we represent its inverse as [latex]f^{-1}(x)[/latex], read as “[latex]f[/latex] inverse of [latex]x[/latex].”

The raised [latex]-1[/latex] is part of the notation. It is not an exponent; it does not imply a power of [latex]-1[/latex]. In other words, [latex]f^{-1}(x)[/latex] does not mean [latex]\frac{1}{f(x)}[/latex] because [latex]\frac{1}{f(x)}[/latex] is the reciprocal of [latex]f[/latex] and not the inverse.
Given a function [latex]f(x)[/latex], we can verify whether some other function [latex]g(x)[/latex] is the inverse of [latex]f(x)[/latex] by checking if both [latex]g(f(x)) = x[/latex] and [latex]f(g(x)) = x[/latex] are true.
[latex]\\[/latex]
For example, [latex]y = 4x[/latex] and [latex]y = \frac{1}{4}x[/latex] are inverse functions.

[latex](f^{-1} \circ f)(x) = f^{-1}(4x) = \frac{1}{4}(4x) = x[/latex]

and

[latex](f \circ f^{-1})(x) = f\left(\frac{1}{4}x\right) = 4\left(\frac{1}{4}x\right) = x[/latex]

A few coordinate pairs from the graph of the function [latex]y = 4x[/latex] are [latex](-2, -8)[/latex], [latex](0, 0)[/latex], and [latex](2, 8)[/latex]. A few coordinate pairs from the graph of the function [latex]y = \frac{1}{4}x[/latex] are [latex](-8, -2)[/latex], [latex](0, 0)[/latex], and [latex](8, 2)[/latex]. If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.

In simpler terms, inverse functions undo each other. If you graph both functions, the coordinates of one function’s graph can be swapped to appear on the graph of its inverse.

inverse function

An inverse function, denoted as [latex]f^{-1}(x)[/latex] reverses the operation of the original function [latex]f(x)[/latex]. The notation[latex]f^{-1}[/latex] is read “[latex]f[/latex] inverse.”

For a function to have an inverse, it must be one-to-one (injective), meaning each output corresponds to exactly one input.

 

Properties of an inverse function

  • Symmetry: The graph of the inverse function is a reflection of the graph of the original function across the line [latex]y = x[/latex]. If the point [latex](a,b)[/latex] lies on the graph of [latex]f(x)[/latex], then the point [latex](b,a)[/latex] lies on the graph of [latex]f^{-1}(x)[/latex].
  • Reversibility: The function and its inverse satisfy the conditions: [latex]f(f^{-1}(x)) = x[/latex] and [latex]f^{-1}(f(x)) = x[/latex]
A function is one-to-one if each output is paired with exactly one input. This means no two different inputs produce the same output. Mathematically, for a function [latex]f(x)[/latex], if [latex]f(a)=f(b)[/latex], then [latex]a=b[/latex].
[latex]\\[/latex]
A one-to-one function passes the horizontal line test—any horizontal line drawn through the graph intersects the graph at most once.
If for a particular one-to-one function [latex]f\left(2\right)=4[/latex] and [latex]f\left(5\right)=12[/latex], what are the corresponding input and output values for the inverse function?

How To: Given two functions [latex]f\left(x\right)[/latex] and [latex]g\left(x\right)[/latex], test whether the functions are inverses of each other.

  1. Determine whether [latex]f\left(g\left(x\right)\right)=x[/latex] and [latex]g\left(f\left(x\right)\right)=x[/latex].
  2. If both statements are true, then [latex]g={f}^{-1}[/latex] and [latex]f={g}^{-1}[/latex]. If either statement is false, then [latex]g\ne {f}^{-1}[/latex] and [latex]f\ne {g}^{-1}[/latex].
If [latex]f\left(x\right)=\dfrac{1}{x+2}[/latex] and [latex]g\left(x\right)=\dfrac{1}{x}-2[/latex], is [latex]g={f}^{-1}?[/latex]

If [latex]f\left(x\right)={x}^{3}[/latex] (the cube function) and [latex]g\left(x\right)=\frac{1}{3}x[/latex], is [latex]g={f}^{-1}?[/latex]