Algebra and Trigonometry, 5th Edition

Restricting a Domain

JUST IN TIME 17

In the case in which the inverse of a function is not a function, the domain of the function can be restricted to allow the inverse to be a function. Let’s consider the function $f(x)=x^2-2$. It is not one-to-one. The graph is shown at left.

Suppose that we had tried to find a formula for the inverse as follows:

$$\begin{array}{rcll}y& =& x^2-2& \mathbf{\text{Replacing }}\mathit{f}\mathbf{(}\mathit{x}\mathbf{)}\mathbf{\text{ with }}\mathit{y}\\ x& =& y^2-2& \mathbf{\text{Interchanging }}\mathit{x}\mathbf{\text{ and }}\mathit{y}\\ x+2& =& y^2\\ \pm \sqrt{x+2}& =& y.& \mathbf{\text{Solving for }}\mathit{y}\end{array}$$

This is not the equation of a function. An input of, say, 2 would yield two outputs, −2 and 2. In such cases, it is convenient to consider “part” of the function by restricting the domain of $f(x)$. For example, if we restrict ...