4 advance functions

4.1 hybrid functions

\[f(x)=\begin{cases} x+2 & x\leq0 \\ -x^2+2 & 0<x\leq2 \\ -2 & x>2 \end{cases}\]

hybrid functions are functions that have different rules for different subsets of their domain

the inverse function \(f^{-1}(x)\) of a function \(f(x)\) is the same as reflecting the the graph \(y=f(x)\) over the line \(y=x\)

to find the inverse of a function \(y=f(x)\) swap all \(x\) and \(y\)

a composite function \(f(g(x))\) or \(f\circ g(x)\) is a compose of a function inside another function

\(\text{range}(g(x))\in \text{domain}(f(x))\)
\(\text{domain}(f\circ g(x))=\text{domain}(g(x))\)
a composite function can exist if the range of the inside function is a subset of the domain of the outer function
the domain of the composite function is the domain of the inside function

an even function are function that are symmetrical about the \(y\) axis

\[f(-x)=f(x)\]

an odd function are function that are rotationally symmetrical about the origin

\[f(-x)=-f(x)\]