Big Idea:

Big Idea 4

An inverse function is a function that "undoes"€ another function; if f(x) maps x to y, then f-1(x) maps y back to x.

1 week

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 = 2x2 - 5 has an inverse with +5,½, and a square root
    • only inverse functions have commutative compositions
       
  • 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 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    

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

Core Resource
A core resource supports multiple days of instruction.
  • Investigating Inverses
    This is a multi-day resource to support students in understanding different methods for determining if two functions are inverses and how each of those methods are related.
    Resource:
    Investigating Inverses

    This is a multi-day resource to support students in understanding different methods for determining if two functions are inverses and how each of those methods are related.  

    All Resources From:
Instructional Routine: Contemplate then Calculate
These tasks are embedded within the instructional routine called Contemplate then Calculate. COMING SOON!
    Instructional Routine: Connecting Representations
    These tasks are embedded within the instructional routine called Connecting Representations.