Precalculus Review

Inverse Functions

A function

is called a one-to-one function if it satisfies

$\displaystyle f(a) \neq f(b)$ whenever $a \neq b$ $\displaystyle .$

The above condition becomes equivalent to the following: “

always implies

.” When a function

is a one-to-one function, the inverse function

can be defined as the function satisfying

$\displaystyle y = f(x)$ if and only if $\displaystyle \quad x = g(y).$

We denote the inverse function of

by the symbol $f^{-1}$ .

Domain of $f^{-1}$ . If is a one-to-one function from the domain to the range , then the inverse function $f^{-1}$ becomes a one-to-one function from the domain to the range .

Horizontal line test. If a function is one-to-one, every “horizontal line” should intersect the graph of in at most one point.

Alternative definition of inverse function. Suppose that is a one-to-one function with domain and range . Then the inverse function $f^{-1}$ can be viewed as the set $W = \{(x, f^{-1}(x)): x \in R\}$ on the rectangular coordinate system. Furthermore, the set is given by

$\displaystyle W = \{(x, f^{-1}(x)): x \in R\} = \{(f(x), x): x \in D\}.$