1\setstretch{1.3} 2\pagenumbering{gobble} 3 4\hypertarget{inverse-functions}{% 5\section{Inverse functions}\label{inverse-functions}} 6 7\hypertarget{functions}{% 8\subsection{Functions}\label{functions}} 9 10\begin{itemize} 11\tightlist 12\item 13 vertical line test 14\item 15 each \(x\) value produces only one \(y\) value 16\end{itemize} 17 18\hypertarget{one-to-one-functions}{% 19\subsection{One to one functions}\label{one-to-one-functions}} 20 21\begin{itemize} 22\tightlist 23\item 24 \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if 25 \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\ 26 \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1, 27 \(x^3\) is) 28\item 29 horizontal line test 30\item 31 if not one to one, it is many to one 32\end{itemize} 33 34\hypertarget{deriving-f-1}{% 35\subsection{\texorpdfstring{Deriving 36\(f^{-1}\)}{Deriving f\^{}\{-1\}}}\label{deriving-f-1}} 37 38\begin{itemize} 39\tightlist 40\item 41 if \(f(g(x)) = x\), then \(g\) is the inverse of \(f\) 42\item 43 reflection across \(y-x\) 44\item 45 \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\) 46\item 47 inverse \(\ne\) inverse \emph{function} (i.e.~inverse must pass 48 vertical line test)\\ 49 \(\implies f^{-1}(x)\) exists \(\iff f(x)\) is one to one 50\item 51 \(f^{-1}(x)=f(x)\) intersections may lie on line \(y=x\) 52\end{itemize} 53 54\hypertarget{requirements-for-showing-working-for-f-1}{% 55\subsubsection{\texorpdfstring{Requirements for showing working for 56\(f^{-1}\)}{Requirements for showing working for f\^{}\{-1\}}}\label{requirements-for-showing-working-for-f-1}} 57 58\begin{enumerate} 59\def\labelenumi{\arabic{enumi}.} 60\tightlist 61\item 62 start with \emph{``let \(y=f(x)\)''} 63\item 64 must state \emph{``take inverse''} for line where \(y\) and \(x\) are 65 swapped 66\item 67 do all working in terms of \(y=\dots\) 68\item 69 for square root, state \(\pm\) solutions then show restricted 70\item 71 for inverse \emph{function}, state in function notation 72\end{enumerate}