1--- 2geometry: a4paper, margin=2cm 3author: Andrew Lorimer 4header-includes: 5- \usepackage{setspace} 6- \usepackage{fancyhdr} 7- \pagestyle{fancy} 8-\fancyhead[LO,LE]{Year 12 Methods} 9-\fancyhead[CO,CE]{Andrew Lorimer} 10--- 11 12\setstretch{1.3} 13\pagenumbering{gobble} 14 15# Inverse functions 16 17## Functions 18 19- vertical line test 20- each $x$ value produces only one $y$ value 21 22## One to one functions 23 24- $f(x)$ is *one to one* if $f(a) \ne f(b)$ if $a, b \in \operatorname{dom}(f)$ and $a \ne b$ 25$\implies$ unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is) 26- horizontal line test 27- if not one to one, it is many to one 28 29## Deriving $f^{-1}$ 30 31- if $f(g(x)) = x$, then $g$ is the inverse of $f$ 32- reflection across $y-x$ 33- $\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}$ 34- inverse $\ne$ inverse *function* (i.e. inverse must pass vertical line test) 35$\implies f^{-1}(x)$ exists $\iff f(x)$ is one to one 36- $f^{-1}(x)=f(x)$ intersections may lie on line $y=x$ 37 38### Requirements for showing working for $f^{-1}$ 39 401. start with *"let $y=f(x)$"* 412. must state *"take inverse"* for line where $y$ and $x$ are swapped 423. do all working in terms of $y=\dots$ 434. for square root, state $\pm$ solutions then show restricted 445. for inverse *function*, state in function notation