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Inverse Functions

The functions $f$ and $g$ are inverse functions if $f\left(g\left(x\right)\right)=x$ for all $x$ in the domain of $g$ and $g\left(f\left(x\right)\right)=x$ for all $x$ in the domain of $f$ .

The inverse of a function $f$ is usually denoted $f^{-1}$  and read “ $f$ inverse.”  (Note that the superscript $-1$ in $f^{-1}$  is not a exponent).

Suppose that two functions are inverses.  If $\left(a,b\right)$ is a point on the graph of the original function, then the point $\left(b,a\right)$ must be a point on the graph of the inverse function.  The graphs are mirror images of each other with respect to the line $y=x$ .

To find the inverse of a function algebraically, interchange the $x$ and $y$ and solve for $y$ .