3.7 Inverse functions  (Page 2/9)

This holds for all $x$ in the domain of $f.$ Informally, this means that inverse functions “undo” each other. However, just as zero does not have a reciprocal , some functions do not have inverses.

Given a function $f(x),$ we can verify whether some other function $g(x)$ is the inverse of $f(x)$ by checking whether either $g(f(x))=x$ or $f(g(x))=x$ is true. We can test whichever equation is more convenient to work with because they are logically equivalent (that is, if one is true, then so is the other.)

For example, $y=4x$ and $y=\frac{1}{4}x$ are inverse functions.

and

A few coordinate pairs from the graph of the function $y=4x$ are (−2, −8), (0, 0), and (2, 8). A few coordinate pairs from the graph of the function $y=\frac{1}{4}x$ are (−8, −2), (0, 0), and (8, 2). If we interchange the input and output of each coordinate pair of a function, the interchanged coordinate pairs would appear on the graph of the inverse function.

Finding domain and range of inverse functions

The outputs of the function $f$ are the inputs to $f^{-1},$ so the range of $f$ is also the domain of $f^{-1}.$ Likewise, because the inputs to $f$ are the outputs of $f^{-1},$ the domain of $f$ is the range of $f^{-1}.$ We can visualize the situation as in [link] .