Often, you need to ‘undo’ what a function did.
For example:
Not all functions can be ‘undone’.
To ‘undo’ means to go from an output back to the input it came from—this only works if the output came from only one place!
That is, in order to ‘undo’ a function, it must satisfy the following
equivalent conditions:
If $\,f\,$ is onetoone, then there exists a unique function $\,f^{1}\,$ (read as ‘$\,f\,$ inverse’) that ‘undoes’ what $\,f\,$ does, as the diagram below illustrates:
There are two mathematical sentences that emerge in this diagram:
$f(x) = y$  $f^{1}(y) = x$ 
‘$\,f\,$ takes $\,x\,$ to $\,y\,$’  ‘$\,f^{1}\,$ takes $\,y\,$ to $\,x\,$’ 
More precisely, the equivalence of these two sentences gives all the following information:
going from $\,y = f(x)\,$ to $\,f{\,}^{1}(y) = x\,$ : let $\,f^{1}\,$ act on both sides 
going from $\,f{\,}^{1}(y) = x\,$ to $\,y = f(x)\,$ : let $\,f\,$ act on both sides 

Start with the equation:  $y = f(x)$  Start with the equation:  $f^{1}(y) = x$  
Let $\,f^{1}\,$ act on both sides:  $f^{1}(y) = f^{1}\bigl(f(x)\bigr)$  Let $\,f\,$ act on both sides:  $f\bigl(f^{1}(y)\bigr) = f(x)$  
Since $\,f\,$ and $\,f^{1}\,$ ‘undo’ each other, they ‘cancel out’ (this is made precise in the next lesson) 
$f^{1}(y) = \underbrace{f^{1}\bigl(f(x)\bigr)}_{= x}$ 
Since $\,f\,$ and $\,f^{1}\,$ ‘undo’ each other, they ‘cancel out’ (this is made precise in the next lesson) 
$\underbrace{f\bigl(f^{1}(y)\bigr)}_{= y} = f(x)$  
This leaves us with:  $f^{1}(y) = x$  This leaves us with:  $y = f(x)$ 
There is some unfortunate notation used for inverse functions, which can lend itself to confusion if you're not careful.
You know from properties of exponents that $\,x^{1}\,$ denotes the multiplicative inverse of $\,x\,$. That is, $\,x^{1} = \frac{1}{x}\,$.
However, when $\,f\,$ is a (onetoone) function,
then $\,f^{1}\,$ does NOT mean $\,\frac{1}{f}\,$!
Instead, $\,f^{1}\,$ is just notation for the inverse of $\,f\,$—the unique function that ‘undoes’ what $\,f\,$ does.
Be careful about this!
Let $\,f(x) = x^3\,$.
Answer the following questions:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
