1# Inverse functions 2 3## Functions 4 5- vertical line test 6- each $x$ value produces only one $y$ value 7 8## One to one functions 9 10- $f(x)$ is *one to one* if $f(a) \ne f(b)$ if $a, b \in \operatorname{dom}(f)$ and $a \ne b$ 11- i.e. unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is) 12- horizontal line test 13- if not one to one, it is many to one 14 15## Inverse functions $f^{-1}$ 16 17- if $f(g(x)) = x$, then $g$ is the inverse of $f$ 18- reflection across $y-x$ 19- $\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}$ 20- inverse $\ne$ inverse *function* (i.e. inverse must pass vertical line test) 21- - $\implies f^{-1}(x)$ exists $\iff f(x)$ is one to one 22- $f^{-1}(x)=f(x)$ intersections may lie on line $y=x$ 23 24Requirements for showing working for $f^{-1}$: 25 26- start with *"let $y=f(x)$"* 27- must state *"take inverse"* for line where $y$ and $x$ are swapped 28- do all working in terms of $y=\dots$ 29- for square root, state $\pm$ solutions then show restricted 30- for inverse *function*, state in function notation