methods / inverse-functions.mdon commit [methods] compile SAC reference (3a41125)
   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