The trig functions all have inverses, but only under special conditions — you have to restrict the domain values. Not all functions have inverses, and not all inverses are easy to determine. Here's a nice method for finding inverses of basic algebraic functions.

## Use algebra to find an inverse function

The most efficient method for finding an inverse function for a given one-to-one function involves the following steps:

1. Replace the function notation name with y.

2. Reverse all the x's and y's (let every x be y and every y be x).

3. Solve the equation for y.

4. Replace y with the function notation for an inverse function.

For example, follow these steps to find the inverse function for

1. Replace the function notation with y.

2. Reverse the x's and y's.

3. Solve for y.

4. Replace y with the inverse function notation.

f–1(x) = (x – 8)3 + 2

Look at how these two functions work. Input 3 into the original function and then get the number 3 back again by putting the output, 9, into the inverse function.

1. Replace the x's with 3 in the function.

2. Replace the x's with 9 in the inverse function.

f–1(9) = (9 – 8)3 + 2 = 13 + 2 = 3

## Use new definitions of functions for inverses

Sometimes you just don't have a nice or convenient algebraic process that will give you an inverse function. Many functions need a special, new rule for their inverse. Here are some examples of these functions:

Function Inverse
f(x) = ex f–1(x) = ln x
g(x) = logax g–1(x) = ax
h(x) = sin x h–1(x) = arcsin x or sin–1 x
k(x) = tan x k–1(x) = tan–1 x or arctan x

If you have a scientific or graphing calculator, you can try out some of these functions and their inverses. Use the function f(x) = ex and its inverse, f–1(x) = ln x, for the following demonstration:

1. In the calculator, use the ex button (often a second function of the calculators) to enter e3.

The input value here is 3. The answer, or output, comes out to be about 20.08553692. This value isn't exact, but it's good for eight decimal places.

2. Now take that answer and use the ln button to find ln 20.08553692.

Input 20.08553692 into the ln function. The answer, or output, that you get this time is 3.