Back To Course

Math 101: College Algebra12 chapters | 94 lessons | 11 flashcard sets

Watch short & fun videos
**Start Your Free Trial Today**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 55,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Luke Winspur*

Luke has taught high school algebra and geometry, college calculus, and has a master's degree in education.

Function composition is the process of putting two or more functions together. This video lesson will explain how this process works and also show you how to evaluate functions that have been composed.

It's really easy for mathematicians to make things seem much harder than they actually are. This often comes down to either confusing vocabulary or confusing notation. While these words or symbols will always have a purpose and will end up making life easier, when you're first learning them it can be hard to keep it all straight.

The topic that this lesson is on, **function composition**, is one of those topics. It can seem complicated at first, so let's start small and ease you into it.

We'll begin by reviewing what function notation is.

Basically, it's just another way of writing an equation. Instead of saying *y* = 4*x* - 1, we can say *f*(*x*) = 4*x* - 1. This notation now gives this function a name, *f*, and allows us to substitute anything we want into it.

Instead of *f*(*x*), what if it was *f*(*w*)? That means *f*(*w*) is just 4*w* - 1.

We don't just have to use symbols, either. How about *f*(6)? Now we just put a 6 in that spot: 4(6) - 1 = 23.

We could even use random shapes if we want! How about *f*(** :)**)? I just plug that smiley face right in, which means

Let's up the difficulty a little bit. Instead of substituting in a single term, what if we tried an expression with multiple terms? Maybe *f*(-2*m*+3)? Just because it's a bigger expression doesn't mean we do anything different. Where there used to be an *x* (or a smiley, or a 6, or a *w*), now I put -2*m* + 3. That gives us this: 4(-2*m* + 3) - 1, which we can then simplify with the distributive property and combining like terms to end up with our answer: -8*m* + 11.

So, as you can see, we can substitute any old thing into a function. So, why not another function? That's exactly what a **composition of functions** is - we take one function and plug it into another one. If we defined another function, let's say *g*(*x*) to be 3*x*^2, we can then evaluate *f*(*g*(*x*)) by doing exactly what we have been doing for the last few minutes and just plug one function into another!

We start with the outside function, *f*: 4 times something - 1, but everywhere that we would normally have put an *x*, we now substitute in the function *g*(*x*). So instead of 4*x* - 1, or 4*w*- 1, or 4 *:)* - 1, we have 4(*g*(*x*)) - 1. But since we know that *g*(*x*) is just 3*x*^2, we can substitute that in as well, which makes *f*(*g*(*x*)) equal to 4(3*x*^2) - 1. Simplifying again gives us our final answer as 12*x*^2 - 1.

And that's it! But composing functions can be difficult because seeing all those letters - *f* and *g* and *x* - can be daunting. Even when you get that part, it can be easy to do the problem backwards and substitute the functions into each other the wrong way. So, let's look at an example or two, and see if we can address those two common mistakes and prevent them from happening to you.

Let's set up some new functions - maybe *r*(*x*) = -*x* + 1 and *s*(*x*) = 2*x* + 5 - and run through the different ways we could compose them.

How about *r*(*s*(*x*))? Well, *r* is the outside function, so we start with that: negative something plus 1. But instead of an *x*, we're substituting in *s*(*x*). That turns what we have, -*x* + 1, into -(2*x* + 5) + 1. Again, distributing and simplifying gives us *r*(*s*(*x*)) = -2*x*- 4.

How about the other way: *s*(*r*(*x*))? This time the outside function is *s*, which means we'll start with 2*x* + 5, but then substitute the *r* function where the *x* used to be. That gives us 2(-*x* + 1) + 5, and our simplified answer is -2*x* + 7.

Notice that we get different answers when we compose the functions in different directions. This means that you've got to be careful to not do them in the wrong way. I limit my mistakes by always starting by writing down the outside function, and only then do I think about the inside one.

There are a few other ways to make these problems slightly more complex. One of those is to compose a function with itself. Maybe *r*(*r*(*x*)): *r* is the outside function, so we start with -*x *+ 1, but then *r* is the inside function as well, so where we saw the *x*, we put another -*x* + 1. That gives us this: -(-*x* + 1) + 1, which simplifies down to just plain *x*.

We can also evaluate a composition of functions at a specific value - maybe like *s*(*s*(3)). We start with the *s* function, 2*x* + 5, substitute in another *s* function, 2(2*x* + 5) + 5, and then substitute a 3 into that (where the *x* used to be), giving us 2(2(3) + 5) + 5. Now, instead of just simplifying, we multiply and add it out. 2 times 3 is 6 plus 5 is 11 times 2 is 22 plus 5 is 27. So, *s*(*s*(3)) is just 27!

Hopefully, this has helped to remove some of the confusing nature of compositions of functions, and show it's simply another way of plugging things into equations. Let's quickly review the highlights.

We can substitute anything we want into a function - variables, shapes, numbers and even other functions!

That's what it means to compose functions - plugging one function into another.

When doing so, begin with the outside function and work your way inside by changing the *x* into whatever new function you are asked to substitute in.

When evaluating a composition of functions at a specific numeric value, do the same process, but then plug that number in where the *x* used to be.

To unlock this lesson you must be a Study.com Member.

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
5 in chapter 7 of the course:

Back To Course

Math 101: College Algebra12 chapters | 94 lessons | 11 flashcard sets

- Functions: Identification, Notation & Practice Problems 9:24
- Transformations: How to Shift Graphs on a Plane 7:12
- What Is Domain and Range in a Function? 8:32
- How to Add, Subtract, Multiply and Divide Functions 6:43
- How to Compose Functions 6:52
- Applying Function Operations Practice Problems 5:17
- Go to Functions

- Trauma Certified Registered Nurse (TCRN): Study Guide
- English 310: Short Stories
- Nurse Entrance Test (NET): Exam Prep & Study Guide
- CCMA Basic Exam: Study Guide & Test Prep
- Personalized Learning in the Classroom
- Diseases of the Central Nervous System
- TOEFL Vocabulary: Words & Practice
- SAT Vocabulary Practice
- Patient Assessments for Trauma Nurses
- Test-Taking Strategies for Reading Comprehension
- Professional Publications in Literacy
- Dyslexia Programs in Texas
- Study.com's Teacher Edition
- Study.com School Plans
- Study.com's Virtual Classrooms
- How to Set Up a Class and Invite Students in Your Study.com Virtual Classroom
- How to View Grades and Export CSVs in Your Study.com Virtual Classroom

- Implied Powers of Congress: Definition & Examples
- Brief History of Germany
- Biological Contamination of Food
- Emerging Technologies in Nursing
- Hubert Humphrey: Presidential Campaign & Platform
- How to Write & Use a Technical Specification Document
- Worcester v. Georgia: Lesson for Kids
- Common Characteristics of Fingerprints
- L.S. Lowry: Quiz & Worksheet for Kids
- Quiz & Worksheet - Korean Folklore & Deities
- Quiz & Worksheet - Graphing & Solving Systems of Inequalities
- Quiz & Worksheet - Data Quality in Healthcare
- Quiz & Worksheet - Calculating Markdown & Discount Pricing
- Developing Presentation Skills Flashcards
- Hypothesis Testing in Statistics Flashcards

- AP Chemistry: Help and Review
- AP English Literature Syllabus Resource & Lesson Plans
- ORELA Physics: Practice & Study Guide
- Management: Skills Development & Training
- Precalculus Algebra for Teachers: Professional Development
- Common Core HS Algebra: Sequences and Series
- History of The European Renaissance
- Quiz & Worksheet - Function & Features of Criminal Law
- Quiz & Worksheet - The Cosmological Principle
- Quiz & Worksheet - Steps in the Phosphorus Cycle
- Quiz & Worksheet - Ionic Chemical Bonds
- Quiz & Worksheet - Correctional Facilities' Custody & Security

- The Atom
- Presidents of the United States: Order & Overview
- Memorial Day Activities
- Causes of the Civil War Lesson Plan
- GRE Math Test & Study Guide
- Reading Comprehension Questions on the LSAT
- MCAT Tips
- Scientific Notation Lesson Plan
- Is the PSAT Hard?
- What is the LSAT?
- Where Can I Find Free SAT Questions?
- Questions to Ask Your College Advisor

Browse by subject