Understanding inverses of functions

The  exercise appears under the Algebra II Math Mission. This exercise practices working with functions in graphical, algebraic and numerical forms.

Types of Problems
There are three types of problems in this exercise:


 * 1) Graph the inverse function: This problem provides a graph of a function. The student is asked to graph the inverse image on the same coordinate axes.Uiof1.png
 * 2) Use the table to evaluate: This problem provides a T-table of inputs and outputs of a function. The student is asked to use the table to evaluate a functional expression involving f, f-inverse, and operations on functions.Uiof2.png
 * 3) Solve for the unknown: This problem is a creative used of inverse functions to create a puzzle with an unknown. The student is expected to use their knowledge of inverse functions to find the unknown that is asked for.Uiof3.png

Strategies
Knowledge of inverse functions, including horizontal line test, are encouraged to ensure success on this exercise.
 * 1) A function is invertible if it passes the horizontal line test, i.e., if all horizontal lines hit the graph at most once.
 * 2) To graph an inverse function, switch c and y-coordinates.
 * 3) The expression $$ f^{-1}(f(x))$$ always evaluates to x.
 * 4) For a function, the output is the y, for an inverse, the output is the x.

Real-life Applications

 * 1) Inverse function "undo" mathematical operations and other functions. They allow the ability to solve equations involving various mathematical functions.
 * 2) An ability to move among inverse function can improve critical thinking and problem solving skills.