Composite and inverse functions

The idea

$fg(x)$ means apply $g$ first, then $f$. An inverse $f^{-1}$ undoes $f$ and exists only if $f$ is one-to-one on its domain. The domain of $f^{-1}$ is the range of $f$, and its graph is the reflection of $f$ in $y = x$.

1. For $fg(x)$, put $g(x)$ into $f$ wherever $x$ appears.
2. For $f^{-1}$: write $y = f(x)$, make $x$ the subject, then swap $x$ and $y$.
3. State the domain of $f^{-1}$ using the range of $f$.

Watch out $f^{-1}(x)$ is the inverse function, not $\frac{1}{f(x)}$.