Evidence of Understanding

**use composition of functions to prove that two functions are inverses**- justify why the output of the composition of
**inverse functions**is the input - describe how operations on the function's domain are counterbalanced in its inverse function
- explain why quadratic and square root functions are inverses
*Example: y = 2x*^{2}*- 5 has an inverse with +5**,**●**½, and a square root*

only inverse functions have **commutative**compositions

- justify why the output of the composition of
**create a function’s inverse using a graph or table of values**- justify two functions are inverses using specific points from the tables of each function
- describe the general shape and characteristics of a given function's inverse
- use multiple representations of a function to help illuminate characteristics of its inverse

- given a function's graph, create a graph or table of values for its inverse function
- determine whether a function is
**one to one**and has an inverse function- explain when domain needs to be restricted to produce an inverse and state the restriction (
**radical**functions and the difference between**square root**and**cube root**)

- explain when domain needs to be restricted to produce an inverse and state the restriction (
- explain why inverse functions
**reflect**over the line y = x

**generate an equation for the inverse of a function and use it to solve problems**- confirm two equations are inverses using compositions of functions or the reflection over y = x
- given a function's graph or table of values, create and justify the function's inverse equation
- use rate of change and other key characteristics to create the inverse equation

- algebraically generate the inverse equation (
*Note: only for simple polynomials- especially linear, quadratic and cubic, or their inverses- radical, cube root, etc.)* - use the inverse equation to describe qualities about the graph or table of f(x) or f
^{-1}(x)- solve square root equations and explain
**extraneous solutions**

- solve square root equations and explain

Develop conceptual understanding:

inverse function, commutative, one to one, radical function, square root, cube root, reflect, extraneous solution

Supporting terms to communicate:

composition, input, output, independent, dependent, domain, range, symmetry, linear, quadratic, cubic