+---
+geometry: margin=2cm
+author: Andrew Lorimer
+header-includes:
+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+---
+
+\setstretch{1.3}
+\pagenumbering{gobble}
+
# Inverse functions
## Functions
## One to one functions
-- $f(x)$ is *one to one* if $f(a) \ne f(b)$ if $a, b \in \operatorname{dom}(f)$ and $a \ne b$
-- i.e. unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is)
+- $f(x)$ is *one to one* if $f(a) \ne f(b)$ if $a, b \in \operatorname{dom}(f)$ and $a \ne b$
+$\implies$ unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is)
- horizontal line test
- if not one to one, it is many to one
-## Inverse functions $f^{-1}$
+## Deriving $f^{-1}$
- if $f(g(x)) = x$, then $g$ is the inverse of $f$
- reflection across $y-x$
- $\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}$
-- inverse $\ne$ inverse *function* (i.e. inverse must pass vertical line test)
-- - $\implies f^{-1}(x)$ exists $\iff f(x)$ is one to one
+- inverse $\ne$ inverse *function* (i.e. inverse must pass vertical line test)
+$\implies f^{-1}(x)$ exists $\iff f(x)$ is one to one
- $f^{-1}(x)=f(x)$ intersections may lie on line $y=x$
-Requirements for showing working for $f^{-1}$:
+### Requirements for showing working for $f^{-1}$
-- start with *"let $y=f(x)$"*
-- must state *"take inverse"* for line where $y$ and $x$ are swapped
-- do all working in terms of $y=\dots$
-- for square root, state $\pm$ solutions then show restricted
-- for inverse *function*, state in function notation
+1. start with *"let $y=f(x)$"*
+2. must state *"take inverse"* for line where $y$ and $x$ are swapped
+3. do all working in terms of $y=\dots$
+4. for square root, state $\pm$ solutions then show restricted
+5. for inverse *function*, state in function notation