Unit 4: Functions and Their Inverses
Lesson Video
Focus Standards
Learning Focus
Additional Resources
Lesson 1: Going to Pieces
A Develop Understanding Task
Defines piecewise functions and provides the opportunity to interpret piecewise functions in a context.
Relate the graph of a function to a story context.
Write a function made up of several functions.
How can I write one function that is made up of pieces of different functions?
Extra Practice Problems
Lesson 2: Piece-Wiser
A Solidify Understanding Task
Solidifies understanding of piecewise functions, as students practice creating and interpreting graphs. Explores the question of whether or not piecewise functions must be continuous.
Graph piecewise functions.
Interpret piecewise functions.
Are all piecewise functions continuous?
Lesson 3: Absolutely Valuable
A Solidify Understanding Task
Introduces the absolute value function as a linear piecewise function. Highlights the key features of the absolute value function.
Relate piecewise functions to absolute value.
Identify features of an absolute value function.
What is the effect of taking the absolute value of a linear function?
How can we think of absolute value as a function?
Lesson 4: Absolutely Transformed
A Practice Understanding Task
Builds fluency in writing equations and in graphing linear absolute value functions by considering transformations of the graph.
Use transformations to graph absolute value functions.
Write the equation that corresponds to the graph of an absolute value function.
How do the graphs of quadratic functions compare to absolute value functions?
Lesson 5: Brutus Bites Back
A Develop Understanding Task
Develops the concept of inverse functions in a linear modeling context using tables, graphs, and equations. Introduces the idea that the domain of the function is the range of the inverse (and vice versa).
Model a relationship between two quantities by using either quantity as the input variable.
How does changing the inputs change the table? The graph? The equation?
Lesson 6: Flipping Ferraris
A Solidify Understanding Task
Extends the concepts of inverse functions in a quadratic modeling context with a focus on domain and range. Discusses when and under what conditions the inverse of a function is also a function. Introduces restricting domains to produce an invertible function from a non-invertible function.
Understand the inverse of a quadratic function.
Determine the relationship between the domain and range of a function and its inverse.
Understand when the inverse of a function is also a function.
Do all functions have inverses? If so, are their inverses always functions?
What type of relation is the inverse of a quadratic function?
Lesson 7: Tracking the Tortoise
A Solidify Understanding Task
Solidifies the concepts of inverse function with an exponential function using graphs and tables. Extends inverse definition to include “undoing” another function. Surfaces informal ideas about logarithms.
Represent the inverse of an exponential function.
Compare the inverse relationship for an exponential function with the inverses of linear and quadratic functions.
Determine if a function is invertible.
What properties of inverse functions for linear and quadratic functions are the same for exponential functions? Can these properties be generalized to any function and its inverse?
Lesson 8: Pulling a Rabbit Out of the Hat
A Solidify Understanding Task
Uses function machines to model functions and their inverses and solidify understanding of the relationship between the two. Focuses on finding inverse functions and verifying that two functions are inverses. Introduces logarithmic notation.
Understand the input-output relationship between a function and its inverse.
Find the inverse of a function.
How can we be sure that two functions are inverses?
How can we find inverse functions?
Lesson 9: Piece Out
A Practice Understanding Task
Practices writing and graphing piecewise and absolute value functions and finding the inverse function.
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NC.M3.F-IF.7
Write and graph piecewise functions efficiently and accurately.
Write and graph absolute value functions in piecewise and absolute value form.
Find inverses and graph inverse functions efficiently and accurately.
How do piecewise functions, absolute value functions, and inverse functions connect?