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Course: Algebra III / Intermediate Algebra
Topic: Functions and Graphs II
Subtopic: Functions III - Composite and Inverse

Overview

This lesson is an extension of on functions introduced in an elementary algebra course (where coverage included the definition of function, function notation, domain and range, and the algebra of functions). There are two concepts of primary importance here. The composition of functions involves putting one function inside another, i.e. taking the function of a function. Be sure that you remember from Elementary Algebra how to combine functions in more basic ways (+, -, x, ÷) first. The inverse of a function means to undo it (as a cube root undoes a cube). Some functions have inverses, some do not. Only those functions that meet certain conditions have inverses (those that do are called one-to-one functions). Be sure you remember from Elementary Algebra how to determine if a relation (given as a list of ordered pairs, mapping, graph, or equation) is a function before learning to determine if a relation is one-to-one.

Notation Caution: f -1(x) means the inverse of f(x) which is the function that "undoes" the function f. f -1(x) does not mean the reciprocal of f(x). In other words, f -1(x) ≠ 1/f(x) even though x-1 = 1/x.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 12.3.1 Correctly use notation for function, composition of functions, and inverse functions
• 12.3.2 Given functions in terms of equations, evaluate compositions and/or inverses
• 12.3.3 Given functions in terms of a list of ordered pairs, evaluate compositions and/or inverses
• 12.3.4 Given functions in terms of graphs, evaluate compositions and/or inverses
• 12.3.5 Given a relation in terms of an equation, a list of ordered pairs, or a graph, determine if it is one-to-one
• 12.3.6 Determine graphically if a function has an inverse
• 12.3.7 Given a one-to-one function, algebraically find its inverse
• 12.3.8 Verify algebraically that two functions are indeed inverses of one another (or determine that they are not)
• 12.3.9 Know the graphical connection between a function and its inverse

Terminology

Define: composition, one-to-one (1:1), inverse vs. reciprocal, inverse function, vertical line test (VLT), horizontal line test (HLT), symmetry about the line y=x

Text Notes

Review concepts from an Elementary Algebra course, where functions were introduced, as needed. Be sure you understand the concept of function, how to determine if a given equation, relation, or graph is a function, and function notation before starting the Intermediate Algebra material that concentrates on composite and inverse functions. This section has lots of new theorems, processes, and terminology. This material forms a foundation on which we will build in the next chapter with the specific exponential and logarithmic functions. Study it well!

Supplementary Resources

Watch these two videos from PatrickJMT.com if you think you would find them helpful:

If you want more video examples, a second excellent source is James Sousa's MathIsPower4U.com. Go to his Algebra II Video Library, scroll down the middle column a bit to the "Determining Composite Functions and Composite Function Values" section and down the middle column a bit more to the "Determining Inverse Functions" section. Choose titles that relate to problems you are needing to study, keeping in mind that some go into College Algebra level material. Stick to the basics here.

Use my Functions and Graphs Checklist as a checklist to be sure that you have learned all that you should!