Every
onetoone function
[beautiful math coming... please be patient]$\,f\,$ has an inverse, denoted by
[beautiful math coming... please be patient]$\,f^{1}\,$, that ‘undoes’ what $\,f\,$
does.
In this lesson and
the next, we look at two common techniques for getting a formula for $\,f^{1}\,$.
The ‘mapping diagram method’ for finding an inverse is discussed below.
It only works when the formula for
[beautiful math coming... please be patient]$\,f\,$ has exactly one appearance of the input variable.
For example,
the function $\,f(x) = 5\color{blue}{x}  4\,$ has only one $\,\color{blue}{x}\,$ in its formula.
However, the function $\displaystyle\,g(x) = \frac{13\color{red}{x}}{5+2\color{red}{x}}\,$ has two appearances of the
variable $\,\color{red}{x}\,$.
The ‘mapping diagram method’ will work for $\,f\,$, but not for $\,g\,$.
This author strongly prefers the ‘mapping diagram method’ because it emphasizes the fact that $\,f\,$ does
something, and $\,f^{1}\,$ undoes it.
The method discussed in the next lesson is more widely applicable, but tends to be quite mechanical—
if you're not careful, you can just ‘go through the motions’ and forget the underlying idea!
THE ‘MAPPING DIAGRAM’ METHOD FOR FINDING $\,f^{1}\,$ 
when the formula for $\,f\,$ contains exactly one $\,x\,$ 
In this example, the ‘mapping diagram’ method for finding the inverse is applied to the function $\,f(x) = 5x  4\,$.

the ‘Mapping Diagram’ method for finding an inverse finding the inverse of $\,f(x) = 5x4\,$: think about what $\,f\,$ does, and then ‘undo’ it, in reverse order 
Want confidence that your inverse is correct?
Verify either one of the following conditions (your choice):
Here's one. The other is similar.
Suppose that $\,f\,$ is onetoone, and
$\,f(f^{1}(y)) = y\,$ for all $\,y\in\text{ran}(f)\,$.
Since $\,f(x)\in\text{ran}(f)\,$ whenever $\,x\in\text{dom}(f)\,$, we have
$\,f\bigl(f^{1}(f(x)\bigr) = f(x)\,$ for all $\,x\in\text{dom}(f)\,$.
But $\,f\,$ is onetoone, so $\,f(z) = f(w)\,$ implies $\,z=w\,$.
Thus, $\,f^{1}(f(x)) = x\,$ for all $\,x\in\text{dom}(f)\,$.
Making this check is also great practice with function composition, so if you have time, it's highly recommended.
Both conditions are checked here for your easy perusal:
Sometimes you have a choice about how you think of ‘undoing’ something.
For example, ‘multiply by 5’ can be undone by ‘divide by 5’.
Or, ‘multiply by 5’ can be undone by ‘multiply by $\frac{1}{5}$’.
As a second example, ‘multiply by $1$’ can be undone by ‘divide by $1$’.
Or, ‘multiply by $1$’ can be undone by ‘multiply by $1$’, since multiplying by $1$ twice
in succession returns you to where you started!
Sometimes, a particular choice can make a formula look a bit simpler.
You don't actually have to draw the mapping diagram to use this method.
Just make a list of what $\,f\,$ does, and how to ‘undo’ each.
Even better—once you get good at this method, you may be able to look at the formula for $\,f\,$ and
write the inverse immediately!
Let's illustrate by finding the inverse of $\,f(x) = 4  \root 3\of{x+1}\,$.
First, rewrite it as $\,f(x) = 1\cdot\root 3\of{x+1} + 4\,$, to make the sequence of operations a bit clearer.
$\,f\,$ does this:  add $\,1\,$  take the cube root  multiply by $\,1\,$  add $\,4\,$ 
undo each:  subtract $\,1\,$  cube  multiply by $\,1\,$  subtract $\,4$ 
Or, you could zip up to http://www.wolframalpha.com and type in something like:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
