Every onetoone function
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$\,f\,$ has an
inverse;
this inverse is denoted by
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$\,f^{1}\,$ and read aloud as ‘$\,f\,$ inverse’.
A function and its inverse ‘undo’ each other: one function does something, the other undoes it.
The purpose of this section is to make this idea precise.
Review: Domain and Range of a Function
The set of allowable inputs for a function $\,f\,$ is called its
domain, and is denoted by $\,\text{dom}(f)\,$.
The set of all possible outputs from a function $\,f\,$ is called its
range, and is denoted by $\,\text{ran}(f)\,$.
That is, let $\,f\,$ act on every possible input—every element of its domain.
The set of resulting outputs is the range of $\,f\,$:
$$\text{ran}(f) = \{ f(x)\ \ x\in\text{dom}(f) \}$$
the Functions $\,f\,$ and $\,f^{1}\,$ ‘Undo’ Each Other
 start with an allowable input for $\,f\,$;
that is, let $\,x\in\text{dom}(f)\,$
 let $\,f\,$ act on $\,x\,$, giving $\,f(x)\,$
 let $\,f^{1}\,$ act on $\,f(x)\,$, giving $\,f^{1}(f(x))\,$
This returns us to where we started, so:
$\,f^{1}(f(x)) = x\,$ for all $\,x\,$ in the domain of $\,f\,$
In other words, the
composite function $\,f^{1}\circ f\,$
is the identity function
on $\,\text{dom}(f)\,$.


 start with an output from $\,f\,$;
that is, let $\,y\in\text{ran}(f)\,$
 let $\,f^{1}\,$ act on $\,y\,$, giving $\,f^{1}(y)\,$
 let $\,f\,$ act on $\,f^{1}(y)\,$, giving $\,f\bigl(f^{1}(y)\bigr)\,$
This returns us to where we started, so:
$\,f\bigl(f^{1}(y)\bigr) = y\,$ for all $\,y\,$ in the range of $\,f\,$
In other words, the
composite function $\,f\circ f^{1}\,$
is the identity function
on $\,\text{ran}(f)\,$.


Input/Output Roles Reversed for $\,f\,$ and $\,f^{1}\,$
The input/output roles for a function and its inverse are reversed—the inputs to one are the outputs from the other.
This fact has some nice consequences:
POINTS ON THE GRAPHS OF $\,f\,$ and $\,f^{1}\,$ HAVE THEIR COORDINATES SWITCHED:
$\,(a,b)\,$ is on the graph of $\,f\,$
if and only if
$\,(b,a)\,$ is on the graph of $\,f^{1}\,$
DOMAIN AND RANGE OF $\,f^{1}\,$:
 the inputs to $\,f^{1}\,$ are the outputs from $\,f\,$:
that is, $\,\text{dom}(f^{1}) = \text{ran}(f)$
 the outputs from $\,f^{1}\,$ are the inputs to $\,f\,$:
that is, $\,\text{ran}(f^{1}) = \text{dom}(f)$
SUMMARY: PROPERTIES OF INVERSE FUNCTIONS
For your convenience, the properties of inverse functions discussed in this and
earlier exercises are summarized below.
PROPERTIES OF INVERSE FUNCTIONS
A function $\,g\,$ has an inverse
if and only if $\,g\,$ is a
onetoone function.
Let $\,f\,$ be a onetoone function.
The inverse of $\,f\,$ is denoted by $\,f^{1}\,$ and read aloud as ‘$\,f\,$ inverse’.
The functions $\,f\,$ and $\,f^{1}\,$ satisfy the following properties:
 $\,f^{1}\,$ is also a onetoone function
 the inverse of $\,f^{1}\,$ is $\,f\,$
 $\,f(x) = y\,$ if and only if $\,f^{1}(y) = x\,$
 $\,f^{1}(f(x)) = x\,$ for all $\,x\in\text{dom}(f)\,$
 $\,f\bigl(f^{1}(y)\bigr) = y\,$ for all $\,y\in\text{ran}(f)\,$
 $(a,b)$ is on the graph of $\,f\,$ if and only if $\,(b,a)\,$ is on the graph of $\,f^{1}\,$
 $\text{dom}(f^{1}) = \text{ran}(f)\,$ and $\text{ran}(f^{1}) = \text{dom}(f)\,$
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
finding inverse functions(when there's only one $\,x\,$ in the formula)