Quick questions on Composite and inverse functions - TCE Mathematics Specialised (Tasmania)

The composite function $f \circ g$ is defined by $(f \circ g)(x) = f(g(x))$: apply $g$ first, then $f$. Order matters, because in general $f(g(x)) \ne g(f(x))$.

The inverse $f^{-1}$ undoes $f$, so $f^{-1}(f(x)) = x$. An inverse function exists only when $f$ is one-to-one, meaning no horizontal line crosses the graph more than once. If $f$ is many-to-one, you must restrict its domain to a piece on which it is one-to-one before an inverse exists.

To find an inverse: write $y = f(x)$, swap $x$ and $y$, then solve for $y$. State the domain of the inverse, which equals the range of the original.

A reliable check is to confirm $f(f^{-1}(x)) = x$. Here $f(f^{-1}(x)) = (\sqrt{x + 4})^2 - 4 = x + 4 - 4 = x$ on the stated domain. If the composition does not return $x$, an algebra error or a domain mistake has crept in.