Modeling, Functions, and Graphs

When you buy a house, your monthly mortgage payment is a function of the size of the loan. The table shows mortgage payments on \(30\)-year loans of various sizes at \(6\)% interest.

For the function \(M = f (L)\text{,}\) the input value is the amount of the loan, and the output is the mortgage payment.

However, when you are shopping for a house, you may think of the mortgage payment as the input variable: If you can afford a certain monthly mortgage payment, how large a loan can you finance? Now the mortgage payment is the input value, and the loan amount is the output. By interchanging the inputs and outputs, we define a new function, \(L = g(M)\text{,}\) shown below.

This new function gives the same information as the original function, \(f\text{,}\) but from a different point of view. We call the function \(g\) the inverse function for \(f\text{.}\)

The elements of the range of \(f\) are used as the input values for \(g\text{,}\) and the output values of \(g\) are the corresponding domain elements of \(f\text{.}\) For example, from the tables you can verify that \(f (200,000) = 1199.10\text{,}\) and \(g(1199.10) = 200,000\text{.}\) In fact, this property defines the inverse function.

If a function is given by a table of values, we can interchange the columns (or rows) of the table to obtain the inverse function. Swapping the columns works because we are really interchanging the input and output variables. If a function is defined by an equation, we can find a formula for its inverse function in the same way: Interchange the roles of the variables in the equation so that the old output variable becomes the new input variable.

If the inverse of a function \(f\) is also a function, we denote the inverse by the symbol \(f^{-1}\text{,}\) read "\(f\) inverse." This notation makes it clear that the two functions are related in a special way. For example, the function \(f (t) = 6 + 2t\) in Example 5.4 has inverse function \(f^{-1}(H) = \dfrac{H -6}{2}\text{.}\)

We can use a graph of a function \(y = f (x)\) to find values of the inverse function \(x = f^{-1}(y)\text{.}\) The figure below shows the graph of \(f(x) = x^3 + 2\text{.}\)

We can always find the inverse of a function simply by interchanging the input and output variables. In the preceding examples, interchanging the variables created a new function. However, the inverse of a function does not always turn out to be a function itself.

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. The graphs of \(f\) and its inverse are shown below. (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.

A function that passes the horizontal line test is called one-to-one, because each input has only one output and each output has only one input. A one-to-one function passes the horizontal line test as well as the vertical line test. With this terminology, we can state the following theorem.

The inverse function \(f^{-1}\) undoes the effect of the function \(f\text{.}\) In Example 5.4, 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 above.

Example 5.23 illustrates the fact that if \(f^{-1}\) is the inverse function for \(f\text{,}\) then \(f\) is also the inverse function for \(f^{-1}\text{.}\)

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 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.