Unit 4 Functions and their Inverses

Lesson 1

Learning Focus

Relate the graph of a function to a story context.

Write a function made up of several functions.

Lesson Summary

In this lesson, we learned about piecewise functions, functions that combine several pieces that each have their own equation into one function. We graphed and wrote equations for piecewise functions. We learned that the equations for each part of the function are called sub-functions, each with their own domain that tells what part of the piecewise function they define.

Lesson 2

Learning Focus

Graph piecewise functions.

Interpret piecewise functions.

Lesson Summary

In this lesson, we graphed piecewise functions and learned that some are discontinuous. We learned how to indicate on a graph whether the point was included in an interval. We also made connections between point-slope form for a line and vertex form for a quadratic function.

Lesson 3

Learning Focus

Relate piecewise functions to absolute value.

Identify features of an absolute value function.

Lesson Summary

In this lesson, we learned about the linear absolute value function. We learned that absolute value functions can be written as piecewise functions or using the operation because they have two distinct parts. We identified the domain and range and graphed the function.

Lesson 4

Learning Focus

Use transformations to graph absolute value functions.

Write the equation that corresponds to the graph of an absolute value function.

Lesson Summary

In this lesson, we learned the quick-graph method for graphing absolute value functions using transformations. We learned to change from absolute value to piecewise form and to identify the vertex and line of symmetry from either form.

Lesson 5

Learning Focus

Model a relationship between two quantities by using either quantity as the input variable.

Lesson Summary

In this lesson, we explored two different ways of viewing a relationship between two quantities. We examined how changing the input quantity changes the output quantity and the relationship between the two functions that are formed using story context, tables, graphs, and equations. We identified features of linear inverse functions that can be seen in each of the representations.

Lesson 6

Learning Focus

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.

Lesson Summary

In this lesson, we examined a quadratic function and its inverse. We found characteristics of inverse functions that are common across function types. We learned that some functions are invertible and that if a function is not invertible, the domain can be restricted to make it invertible.

Lesson 7

Learning Focus

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.

Lesson Summary

In this lesson, we modeled the inverse of an exponential function to determine its features. We learned that this type of function is called a logarithmic function, which we will learn more about in Unit 5. We also discussed a way to describe the input-output relationship of inverse functions using mathematical notation.

Lesson 8

Learning Focus

Understand the input-output relationship between a function and its inverse.

Find the inverse of a function.

Lesson Summary

In this lesson, we learned that the equation of the inverse function has the inverse operations in the reverse order of the original function. Using this idea, we learned a method for finding the inverse of a function if the function is invertible or the domain has been restricted to make it invertible.

Lesson 9

Learning Focus

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.

Lesson Summary

In this lesson, we worked on becoming more fluent with piecewise functions, absolute value functions, and inverse functions. We learned to be more efficient in our work and to refine the details so that it is entirely correct.