Key ideas

Quadratic functions are not one-to-one on all $\mathbb{R}$.

To find an inverse, restrict the domain to one side of the vertex (often $x\ge h$ or $x\le h$), then:

• Write $y=f(x)$
• Swap $x$ and $y$
• Solve for $y$ (choose the correct branch)

The domain of $f^{-1}$ equals the range of $f$, and the range of $f^{-1}$ equals the domain of $f$.