Analyze invertible and non-invertible functions

The  exercise appears under the Algebra II Math Mission. This exercise practices determining whether a given function is invertible or not. If it isn't, students find the necessary changes to make in order to make the function invertible.

Types of problems
There is one type of problem in this exercise:
 * 1) Build the mapping diagram for $$f$$ by dragging the endpoints of the segments in the graph below so that they pair each domain element with its correct range element: This problem explains the values of $$f$$ and has a table with its inputs and outputs. Students are then asked to find the values of the functions multiplied and then evaluate why or why not the functions are inverses.

Strategies
Basic knowledge of finding inverses of functions is required for this exercise.
 * 1) Every linear function's inverse is also linear.
 * 2) Functions can only have one unique output for each input.
 * 3) By the definition of inverse functions, $$f^{-1}(x)=y$$ if and only if $$f(y)=x$$.

Real-life Applications

 * 1) Money as a function of time. One never has more than one amount of money at any time because they can always add everything to give one total amount. By understanding how their money changes over time, they can plan to spend their money sensibly. Businesses find it very useful to plot the graph of their money over time so that they can see when they are spending too much.
 * 2) Temperature as a function of various factors. Temperature is a very complicated function because it has so many inputs, including: the time of day, the season, the amount of clouds in the sky, the strength of the wind, and many more. But the important thing is that there is only one temperature output when they measure it in a specific place.
 * 3) Location as a function of time. One can never be in two places at the same time. If they were to plot the graphs of where two people are as a function of time, the place where the lines cross means that the two people meet each other at that time. This idea is used in logistics, an area of mathematics that tries to plan where people and items are for businesses.