Section 4.1 Inverse Functions

In Example 2.25, we used a graph of \(h\) to read values of \(h^{-1}\text{.}\) But we can also plot the graph of \(h^{-1}\) itself. Because \(C\) is the input variable for \(h^{-1}\text{,}\) we plot \(C\) on the horizontal axis and \(F\) on the vertical axis. To find some points on the graph of \(h^{-1}\text{,}\) we interchange the coordinates of points on the graph of \(h\text{.}\) The graph of \(h^{-1}\) is shown in Figure 4.1.

We can always find the inverse of a function \(y=f(x) \) simply by solving for \(x \) thus interchanging the role of the input and output variables. In the preceding examples, this process created a new function. However, this process does not always lead to be a function.

For example, to find the inverse of \(y = f (x) = x^2\text{,}\) we solve for \(x\) to get \(x = \pm\sqrt{y}\text{.}\) When we regard \(y\) as the input and \(x\) as the output, the relationship does not describe a function. This can be seen as plugging in \(y=4 \text{,}\) for example, gives two outputs \(x=2 \) and \(x=-2\text{.}\) The graphs of \(f\) and its inverse are shown in Figure 4.6. (Note that for the graph of the inverse, we plot \(y\) on the horizontal axis and \(x\) on the vertical axis.) Because the graph of the inverse does not pass the vertical line test, it is not a function.

For many applications, it is important to know whether or not the inverse of \(f\) is a function. This can be determined from the graph of \(f\text{.}\) When we interchange the roles of the input and output variables, horizontal lines of the form \(y = k\) become vertical lines.

Thus, if the graph of the inverse is going to pass the vertical line test, the graph of the original function must pass the horizontal line test , namely, that no horizontal line should intersect the graph in more than one point. Notice that the graph of \(f(x) = x^2\) does not pass the horizontal line test, so we would not expect its inverse to be a function.

We have been talking about how to tell if the inverse of a function is also a function, but in practice this is not the language typically used. Usually we ask this same question in the form "Is the function invertible?" The following definition explains this relationship:

The inverse function \(f^{-1}\) undoes the effect of the function \(f\text{.}\) In Example 2.16, the function \(f(t) = 6 + 2t\) multiplies the input by \(2\) and then adds \(6\) to the result. The inverse function \(f^{-1}(H) = \dfrac{H -6}{2}\) undoes those operations in reverse order: It subtracts \(6\) from the input and then divides the result by \(2\text{.}\)

We return to the original input value, \(5\text{,}\) as illustrated in Figure 4.11.

Example 4.12 illustrates the fact that if \(f^{-1}\) is the inverse function for \(f\text{,}\) then \(f\) is also the inverse function for \(f^{-1}\text{.}\)

Example 4.13 illustrates finding an inverse of a function. The general process is to move operations (or undo them) so the input variable (\(x\)) is what is alone on one side of the equation.

So far we have been careful to keep track of the input and output variables when we work with inverse functions. This is important when we are dealing with applications; the names of the variables are usually chosen because they have a meaning in the context of the application, and it would be confusing to change them.

However, we can also study inverse functions purely as mathematical objects. There is a relationship between the graph of a function and the graph of its inverse that is easier to see if we plot them both on the same set of axes.

Now, for every point \((a, b)\) on the graph of \(f\text{,}\) the point \((b, a)\) is on the graph of the inverse function. Observe in Figure 4.14 that the points \((a, b)\) and \((b, a)\) are always located symmetrically across the line \(y=x\text{.}\) The graphs are symmetric about the line \(y=x\), which means that if we were to place a mirror along the line \(y=x\text{,}\) each graph would be the reflection of the other.