関数：合成と逆関数

The domain is the set of permitted inputs $x$; the range is the set of values that actually come out. For example $f(x)=\sqrt{x-2}$ needs the expression under the root to be non-negative, so the domain is $x\ge 2$, and since the output is non-negative the range is $f(x)\ge 0$. Problems often impose a domain, and that restriction directly affects the range and whether an inverse exists.

An inverse exists only when the function is one-to-one — that is, different inputs always give different outputs. Graphically, every horizontal line meets the curve at most once (the horizontal line test). The function $f(x)=x^2$ is not one-to-one overall, but restricting its domain to $x\ge 0$ makes it one-to-one, so it then has the inverse $\sqrt{x}$.

The composite $fg$ means 'do $g$ first, then $f$', computed as $fg(x)=f(g(x))$. Work from the inner function $g$ outwards to $f$. In general $fg\ne gf$: swapping the order changes the result. Also, for $fg$ to be defined, the range of $g$ must lie within the domain of $f$.

To find an inverse: set $y=f(x)$, swap $x$ and $y$, then make $y$ the subject. A key property: the domain of $f^{-1}$ is the range of $f$, and the range of $f^{-1}$ is the domain of $f$ — domain and range swap over. Moreover the graphs of $y=f(x)$ and $y=f^{-1}(x)$ are mirror images of each other in the line $y=x$.