Inverse Functions

• A function $g$ is the inverse of the function $f$ if $f(g(x))=x$ for each $x$ in the domain of $g$ , and $g(f(x))=x$ for each $x$ in the domain of $f.$ The function $g$ is denoted $f^{-1}.$
• If $g$ is the inverse of $f$ , then $f$ is the inverse of $g.$
• The domain of $f^{-1}$ is equal to the range of $f$ , and the range of $f^{-1}$ is equal to the domain of $f.$
• The graph of $f$ contains the point $(a , b)$ if and only if the graph of $f^{-1}$ contains the point $(b , a).$ Inverse functions are symmetric about the line $y=x.$
• A function possesses an inverse if and only if it is one-to-one.

Finding the Inverse of a Function:

• Determine whether the function $y=f(x)$ has an inverse. It has an inverse if and only if it is one-to-one.
• Solve for $x$ as a function of $y.$
• Interchange $x$ and $y.$ The result is $y=f^{-1}(x).$
• Define the domain of $f^{-1}$ to be the range of $f.$
• Verify that $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x.$