methods / inverse-functions.mdon commit Merge branch 'master' of ssh://charles/tank/andrew/school/notes (9b4ce53)
   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