Recall that a
function can be viewed as a ‘box’:
you drop an input in the top, the function does something to the input, and a unique output drops out the bottom.
In this lesson, we're going to try and use a function box ‘backwards’.
That is, we'll pick up a number from the output pile,
put it in the box ‘backwards’,
and try to see what input it came from.
Sometimes this works out nicely, and sometimes it doesn't!
Consider the squaring function, $\,f(x) = x^2\,$.
Now, pick up the number $\,4\,$ from the output pile. 

Consider the cubing function, $\,f(x) = x^3\,$.
Pick up the number $\,8\,$ from the output pile. 

A function box can only be used ‘backwards’ when every output has exactly one corresponding input!
Imagine sweeping a horizontal line from top to bottom (or bottom to top) through the graph of a function.
At each location, the horizontal line hits the $\,y\,$axis
(but not necessarily the graph of the function!)
in a unique point;
you're ‘testing’ each output to see how many corresponding input(s) it has.
If a horizontal line ever hits the graph at more than one point (as in the squaring function),
then there exists an output with more than one input.
For the squaring function, the horizontal line at height $\,4\,$ hits the graph at two points:
$\ x=2\ $ and $\ x=2\ $.
In this case, we say that the graph ‘fails the horizontal line test’.
If a horizontal line always intersects the graph at only one point,
then every output has only one input.
In this case, we say that the graph ‘passes the horizontal line test’.
For example, the graph of the cubing function passes the horizontal line test.
So—some functions are ‘nicer’ than others,
with respect to using the function box ‘backwards’!
Which are the ‘nice’ ones?
The ones with graphs that pass a horizontal line test!
(Note: Every function already passes a vertical line test.)
For these ‘nice’ functions, there is a beautiful relationship between the inputs and outputs:
since it's a function, each input has exactly one output
(that is, the graph passes a vertical line test);
since it additionally passes a horizontal line test,
each output has exactly one input.
Put together, it is as if the inputs and outputs are connected with strings!
Pick up any input, follow the string to its unique corresponding output.
Pick up any output, follow the string back to its unique corresponding input.
There is a onetoone correspondence between the inputs and outputs.
An input uniquely determines an output,
and an output uniquely determines an input.
In the next lesson, we'll give these ‘nice’ functions a special name:
onetoone functions.
Soon after, we'll see that onetoone functions have inverses that ‘undo’
what the function does,
and we'll study techiques for finding inverses.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
