Form composite functions and find inverse functions, attending to domain and range and the condition for an inverse to exist.

What this dot point is asking

You need to form composite functions with correct domains, determine when an inverse function exists, find inverse functions, and relate a function and its inverse graphically.

Composite functions

The composite $(f \circ g)(x) = f(g(x))$ applies $g$ first and feeds the result into $f$. Order matters: in general $f \circ g \ne g \circ f$.

For example, if $f(x) = \sqrt{x}$ and $g(x) = x - 4$, then $(f \circ g)(x) = \sqrt{x - 4}$, defined only for $x \ge 4$, even though $g$ itself is defined for all real $x$.

When does an inverse exist?

An inverse function $f^{-1}$ reverses $f$: if $f(a) = b$ then $f^{-1}(b) = a$. For this reversal to be a function, each output must come from exactly one input.

Restricting the domain can create an inverse: $f(x) = x^2$ on $x \ge 0$ is one-to-one and has inverse $f^{-1}(x) = \sqrt{x}$.

Finding an inverse algebraically

The mechanical recipe: write $y = f(x)$, swap $x$ and $y$, then solve for $y$. The domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$.

The graphical relationship

The graph of $f^{-1}$ is the reflection of the graph of $f$ in the line $y = x$. Consequently a point $(a, b)$ on $f$ corresponds to $(b, a)$ on $f^{-1}$, and any intersection of $f$ with its inverse lies on the line $y = x$ (for increasing functions). Composing a function with its inverse returns the input:

each on the appropriate domain.