Inverse Functions

When does an inverse exist?

A function $f$ has an inverse exactly when it is one-to-one: no output value is hit twice. Graphically this is the horizontal line test; analytically, on the AP exam, the working criterion is that $f$ is strictly monotonic (always increasing or always decreasing) on the interval in question. A function that turns around — like $x^2$ on all of $\mathbb{R}$ — must first be restricted to a monotonic piece before an inverse can be defined.

The inverse undoes $f$: $f^{-1}(b)=a \iff f(a)=b$. Three consequences carry almost every exam question:

• Domain and range swap: the domain of $f^{-1}$ is the range of $f$, and vice versa.
• The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y=x$ — every point $(a,b)$ becomes $(b,a)$.
• Compositions cancel: $f\left(f^{-1}(x)\right)=x$ on the domain of $f^{-1}$, and $f^{-1}\left(f(x)\right)=x$ on the domain of $f$.

To find a formula: write $y=f(x)$, swap the roles of $x$ and $y$, then solve for $y$. The algebra is routine; stating the correct domain afterwards is where the points hide.