From be2546199cdcce32656eca763c33808120e9bb39 Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Tue, 9 Apr 2019 14:24:03 +1000
Subject: [PATCH] [methods] start inverse functions

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 methods/inverse-functions.md | 30 ++++++++++++++++++++++++++++++
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 create mode 100644 methods/inverse-functions.md

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+# Inverse functions
+
+## Functions
+
+- vertical line test
+- each $x$ value produces only one $y$ value
+
+## 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)
+- horizontal line test
+- if not one to one, it is many to one
+
+## Inverse functions $f^{-1}$
+
+- 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
+- $f^{-1}(x)=f(x)$ intersections may lie on line $y=x$
+
+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
-- 
2.43.2