Inverse Functions: Learn It 3

Graphing Inverse Functions

Let’s consider the relationship between the graph of a function [latex]f[/latex] and the graph of its inverse.

Consider the graph of [latex]f[/latex] shown in Figure 9(a) and a point [latex](a,b)[/latex] on the graph. 

An image of two graphs. The first graph is of “y = f(x)”, which is a curved increasing function, that increases at a faster rate as x increases. The point (a, b) is on the graph of the function in the first quadrant. The second graph also graphs “y = f(x)” with the point (a, b), but also graphs the function “y = f inverse (x)”, an increasing curved function, that increases at a slower rate as x increases. This function includes the point (b, a). In addition to the two functions, there is a diagonal dotted line potted with the equation “y =x”, which shows that “f(x)” and “f inverse (x)” are mirror images about the line “y =x”.
Figure 9. (a) The graph of this function [latex]f[/latex] shows point [latex](a,b)[/latex] on the graph of [latex]f[/latex]. (b) Since [latex](a,b)[/latex] is on the graph of [latex]f[/latex], the point [latex](b,a)[/latex] is on the graph of [latex]f^{-1}[/latex]. The graph of [latex]f^{-1}[/latex] is a reflection of the graph of [latex]f[/latex] about the line [latex]y=x[/latex].

Since [latex]b=f(a)[/latex], then [latex]f^{-1}(b)=a[/latex]. Therefore, when we graph [latex]f^{-1}[/latex], the point [latex](b,a)[/latex] is on the graph. As a result, the graph of [latex]f^{-1}[/latex] is a reflection of the graph of [latex]f[/latex] about the line [latex]y=x[/latex].

How to: Graph the Inverse of a Function

  1. Plot the Function: Graph the original function [latex]f(x)[/latex] and plot a few key points.
  2. Reflect Over Line: Reflect these points over the line [latex]y=x[/latex] to find the corresponding points on [latex]f^{-1}[/latex].
  3. Draw the Inverse: Connect these reflected points to graph the inverse function.
  4. Check: Ensure that each point [latex](a,b)[/latex] on the original function corresponds to the point [latex](b,a)[/latex] on the inverse function.
  5. Line of Symmetry: The line [latex]y=x[/latex] should act as a line of symmetry between the function and its inverse.

For the graph of [latex]f[/latex] in the following image, sketch a graph of [latex]f^{-1}[/latex] by sketching the line [latex]y=x[/latex] and using symmetry. Identify the domain and range of [latex]f^{-1}[/latex].

An image of a graph. The x axis runs from -2 to 2 and the y axis runs from 0 to 2. The graph is of the function “f(x) = square root of (x +2)”, an increasing curved function. The function starts at the point (-2, 0). The x intercept is at (-2, 0) and the y intercept is at the approximate point (0, 1.4).
Figure 10. Graph of [latex]f(x)[/latex].


Restricting Domains

As we have seen, [latex]f(x)=x^2[/latex] does not have an inverse function because it is not one-to-one. However, we can choose a subset of the domain of [latex]f[/latex] such that the function is one-to-one. This subset is called a restricted domain.

By restricting the domain of [latex]f[/latex], we can define a new function [latex]g[/latex] such that the domain of [latex]g[/latex] is the restricted domain of [latex]f[/latex] and [latex]g(x)=f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]g[/latex]. Then we can define an inverse function for [latex]g[/latex] on that domain.

For example, since [latex]f(x)=x^2[/latex] is one-to-one on the interval [latex][0,\infty)[/latex], we can define a new function [latex]g[/latex] such that the domain of [latex]g[/latex] is [latex][0,\infty)[/latex] and [latex]g(x)=x^2[/latex] for all [latex]x[/latex] in its domain. Since [latex]g[/latex] is a one-to-one function, it has an inverse function, given by the formula [latex]g^{-1}(x)=\sqrt{x}[/latex].

On the other hand, the function [latex]f(x)=x^2[/latex] is also one-to-one on the domain [latex](−\infty,0][/latex]. Therefore, we could also define a new function [latex]h[/latex] such that the domain of [latex]h[/latex] is [latex](−\infty,0][/latex] and [latex]h(x)=x^2[/latex] for all [latex]x[/latex] in the domain of [latex]h[/latex]. Then [latex]h[/latex] is a one-to-one function and must also have an inverse. Its inverse is given by the formula [latex]h^{-1}(x)=−\sqrt{x}[/latex] (Figure 13).

An image of two graphs. Both graphs have an x axis that runs from -2 to 5 and a y axis that runs from -2 to 5. The first graph is of two functions. The first function is “g(x) = x squared”, an increasing curved function that starts at the point (0, 0). This function increases at a faster rate for larger values of x. The second function is “g inverse (x) = square root of x”, an increasing curved function that starts at the point (0, 0). This function increases at a slower rate for larger values of x. The first function is “h(x) = x squared”, a decreasing curved function that ends at the point (0, 0). This function decreases at a slower rate for larger values of x. The second function is “h inverse (x) = -(square root of x)”, an increasing curved function that starts at the point (0, 0). This function decreases at a slower rate for larger values of x. In addition to the two functions, there is a diagonal dotted line potted with the equation “y =x”, which shows that “f(x)” and “f inverse (x)” are mirror images about the line “y =x”.
Figure 13. (a) For [latex]g(x)=x^2[/latex] restricted to [latex][0,\infty), \, g^{-1}(x)=\sqrt{x}[/latex]. (b) For [latex]h(x)=x^2[/latex] restricted to [latex](−\infty,0], \, h^{-1}(x)=−\sqrt{x}[/latex].

restricted domain

Some functions don’t have inverses over their full domains because they’re not one-to-one. By restricting the domain, we ensure the function is one-to-one. Once the domain is restricted, we can define an inverse.

Consider the function [latex]f(x)=(x+1)^2[/latex].

  1. Sketch the graph of [latex]f[/latex] and use the horizontal line test to show that [latex]f[/latex] is not one-to-one.
  2. Show that [latex]f[/latex] is one-to-one on the restricted domain [latex][-1,\infty)[/latex]. Determine the domain and range for the inverse of [latex]f[/latex] on this restricted domain and find a formula for [latex]f^{-1}[/latex].