Algebra and Trigonometry, 5th Edition

Finding Formulas for Inverses

Suppose that a function is described by a formula. If it has an inverse that is a function, we proceed as follows to find a formula for $f^{-1}$.

Obtaining a Formula for an Inverse

If a function f is one-to-one, a formula for its inverse can generally be found as follows:

1. Replace $f(x)$ with y.

2. Interchange x and y.

3. Solve for y.

4. Replace y with $f^{-1}(x)$.

Example 6

Determine whether the function $f(x)=2x-3$ is one-to-one, and if it is, find a formula for $f^{-1}(x)$.

Solution

The graph of f is shown at left. It passes the horizontal-line test. Thus it is one-to-one and its inverse is a function. We also proved that f is one-to-one in Example 3. We find a formula for $f^{-1}(x)$.

$$\begin{array}{lrcl}\mathbf{1.}\mathrm{Replace}f(x)\mathrm{with}y:& y& =& 2x-3\\ \mathbf{2.}\mathrm{Interchange}x\mathrm{and}\end{array}$$