Inverse Functions

You have used inverse functions since your first days of algebra to cancel things out. Now that you possess a higher degree of mathematical proficiency, you can better explore why and how they work.

What is an inverse function?

The inverse function for f( x), labeled f −1( x) (which is read “ f inverse of x”), contains the same domain and range elements as the original function, f( x). However, the sets are switched. In other words, the domain of f( x) is the range of f −1( x), and vice versa. In fact, for every ordered pair ( a, b) belonging to f( x), there is a corresponding ordered pair ( b, a) that belongs to f −1( x). For example, consider this function, g:



The inverse function is the set of all ordered pairs reversed:



Only one‐to‐one functions possess inverse functions. Because these functions have range elements that correspond to only one domain element each, there's no danger that their inverses will not be functions. The horizontal line test is a quick way to determine whether a graph is that of a one‐to‐one function. It works just like the vertical line test: If an arbitrary horizontal line can be drawn across the graph of f( x) and it intersects f in more than one place, then f cannot be a one‐to‐one function.

Inverse functions have the unique property that, when composed with their original functions, both functions cancel out. Mathematically, this means that

Graphs of inverse functions

Since functions and inverse functions contain the same numbers in their ordered pair, just in reverse order, their graphs will be reflections of one another across the line y = x, as shown in Figure 1.

 Figure 1 Inverse functions are symmetric about the line y = x.


Finding inverse functions

To find the inverse function for a one‐to‐one function, follow these steps:

1. Rewrite the function using y instead of f( x).

2. Switch the x and y variables; leave everything else alone.

3. Solve the new equation for y.

4. Replace the y with f −1( x).

5. Make sure that your resulting inverse function is one‐to‐one. If it isn't, restrict the domain to pass the horizontal line test.

Example 1: If , find f −1( x).

Follow the five steps previously listed, beginning with rewriting f( x) as y:



Note the restriction x ≥ 0 for f −1( x). Without this restriction, f −1( x) would not pass the horizontal line test. It obviously must be one‐to‐one, since it must possess an inverse of f( x). You should use that portion of the graph because it is the reflection of f( x) across the line y = x, unlike the portion on x < 0.