Every onetoone function $\,f\,$ has an inverse,
denoted by $\,f^{1}\,,$
that ‘undoes’ what $\,f\,$ does.
In this lesson and the previous one,
we look at two common techniques for getting a formula for $\,f^{1}\,.$
This author strongly prefers the mapping diagram method of the previous lesson,
because it emphasizes the fact that $\,f\,$ does something, and $\,f^{1}\,$ undoes it.
That method, however, only works when the formula for $\,f\,$ contains exactly one appearance of the input variable.
The method discussed in this lesson,
dubbed the ‘Switch Input/Output Names’ method,
is more widely applicable.
However, it tends to be quite mechanical—if you're not careful, you can just ‘go through the motions’ and forget the underlying idea!
The input/output roles for a function and its inverse are switched—the inputs to one are the outputs from the other.
If a function $\,f\,$ takes $\,x\,$ to $\,y\,,$ then $\,f^{1}\,$ takes $\,y\,$ back to $\,x\,.$
In other words, if $\,y = f(x)\,,$ then $\,f^{1}(y) = x\,.$
This is the reason we ‘switch the names’ in the method discussed next!
In this example, the ‘switch input/output names’ method for finding the inverse
is applied to the function $\displaystyle\,f(x) = \frac{13x}{5+2x}\,.$
Note that the mapping diagram method cannot be used for this function,
since it contains two appearances of the input variable $\,x\,.$
$\displaystyle x = \frac{13y}{5+2y}$  you must get all the variables $\,y\,$ ‘upstairs’, on the same side of the equation 
$x(5+2y) = 13y$  start by clearing fractions 
$5x + 2xy = 1  3y$  multiply out 
$2xy + 3y = 1  5x$  rearrange: get all terms containing $\,y\,$ on the same side; move other terms to the other side 
$y(2x + 3) = 1  5x$  factor out $\,y\,$ 
$\displaystyle y = \frac{15x}{2x+3}$  solve for $\,y\,$ 
It's fantastic practice to check that $\,f(f^{1}(x)) = x\,$ and $\,f^{1}(f(x)) = x\,.$
Along the way you end up with ‘complex fractions’—fractions within fractions.
Note the multiplybyone technique used to turn these complex fractions into ‘simple’ fractions!
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
