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.

Adding, subtracting, multiplying and dividing functions is about as simple as substituting in expressions and then just doing whichever operation it asks you to do. Check out this video lesson to see some examples of this and learn just how easy it is!

I'm a pretty big sports fan, but I've always been bummed out by how expensive it is to buy gear from my favorite teams. But when I recently moved to Minneapolis, I made some friends that have decided to do something about it! They just opened their own t-shirt company called Tinyapolis that sells t-shirts for the popular teams here in Minnesota.

But when you own your own business, you want to be sure that you're going to be able to make money. So before they took the plunge and bought all the supplies to begin making their shirts, they figured out what their **revenue function**, or **r(x)**, would be. This is the function that would tell them how much money they would make from selling *x* t-shirts. But it's just as important to know what the **cost function**, or **c(x)**, would be. This would tell them how much money they would have to spend in order to make *x* t-shirts.

After doing a little research, they came up with the revenue and cost functions seen here: r(*x*) = 20*x* and c(*x*) = *x*^2 - 1100*x* + 1200. But separate, these two functions don't tell the whole story. What is most important is, after it is all said and done and the t-shirts have been made and sold, did they make money or lose money?

That's where the **profit function**, or **p(x)**, comes in. The profit function would tell my friends whether they would make more money from selling the shirts than it would cost them to make them. This means that the profit function is simply the revenue function minus the cost function. If it costs more to make *x* t-shirts than they make from selling them, they'll have negative profit. But if they make more from the sales than they spend producing the shirts, they'll be in good shape!

So what does this profit function actually look like? Well, all we really have to do is substitute in what we already know the revenue and cost functions are and then simplify. First we'll go ahead and distribute the negative sign to the *x*^2, the -1100*x* and the 1200. Then we combine like terms by grouping together the 20*x* and the 1100*x*, and we end up with our profit function as this: p(*x*) = -*x*^2 + 1120*x* - 1200. Whoa, if those numbers are correct, they're going to be rolling in it. I hope they spread the wealth!

This was an example of a **function operation** - specifically, subtraction. But we can do all the major operations on functions, such as addition, multiplication and division. All of these different operations simply require you to substitute in what you know the function is and go from there, which really isn't too bad, but there are a few reasons that these problems can get tricky.

First, just the function notation itself often confuses people into thinking it's more difficult than it actually is. Secondly, there is a good amount of prerequisite knowledge you need to know in order to fully solve function operation questions. This is because each operation will end up asking you to do something a little different. For example, when we subtracted functions just a few seconds ago, we were required to combine like terms. But when you multiply functions, you'll often have to remember how to multiply polynomials with FOIL or the area method. But that means as long as you're comfortable with function notation and have a solid algebra background, there isn't anything to it.

Let's take a look at a different example that will ask us to use a different operation, maybe multiplication like I just mentioned. If f(*x*) = *x*^2 + 2*x* - 5 and g(*x*) = 3*x* - 1, then what is f(*x*) * g(*x*)?

Let's avoid the first pitfall and not let all this function notation freak us out. All we're being asked to do is multiply the two functions, so we can substitute in those expressions they listed for us like this. At this point, it's simply a matter of using our algebra skills to simplify the expression. Because we're multiplying polynomials, that means FOIL-type multiplying. Basically, we need to multiply each term from this front expression with each term in the second one.

I like using the area method to organize my work for multiplying polynomials that have more than two terms. That requires us to make a chart that has dimensions equal to the number of terms in our two polynomials - in this case, three by two. Next, we label the top and the side of the chart with our two polynomials. So the top, which is divided into three sections, gets f(*x*) (*x*^2, 2*x* and -5), while the left of the chart gets g(*x*) (3*x* and -1). Now we fill in the chart by multiplying the terms to the left and above each box. For example, the first box will be *x*^2 * 3*x* = 3*x*^3. Continuing this process would look like this, and now we just combine whatever like terms we have to find our final answer is 3*x*^3 + 5*x*^2 - 17*x* + 5.

Dividing functions is a very similar process. We won't have to multiply with the area method, but we'll still substitute in expressions and simplify whatever we're left with. Let's try this example: given g(*x*) = 4*x*^5 and h(*x*) = 8*x*^2, what is g(*x*) / h(*x*)?

Don't worry about the notation; let's just substitute in what we know g and h are. That leaves us with this: 4*x*^5 / 8*x*^2. Now we just need to simplify our expression, this time with our exponent properties.

We can either use the quotient of powers property, which tells us to subtract the exponents, or we can simply write out the *x*s we've got and cancel out any that are on the top and the bottom. Either way, we should end up with only three *x*s left in the numerator. Now we can simplify the numbers using middle school simplifying of fractions skills to say that 4/8 is the same as 1/2, and we've got our answer: *x*^3/2!

We can add, subtract, multiply or divide functions by substituting in what we know the function is and then simplifying the expression. These things are called function operations. The key to function operations is to not let the function notation get you off track. Substitute in what you know the functions are, and then use whatever algebra knowledge you have to simplify the expression.

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
4 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
- Inverse Functions 6:05
- Applying Function Operations Practice Problems 5:17
- Go to Functions

- C (ASCP) Technologist in Chemistry: Study Guide & Exam Prep
- MLT (ASCP) Medical Laboratory Technician: Study Guide & Exam Prep
- Pennsylvania Grades 4-8 Core Assessment - Mathematics & Science (5155): Study Guide & Test Prep
- Literary Elements Lesson Plans & Resources
- Pennsylvania Grades 4-8 Core Assessment - English Language Arts & Social Studies (5154): Study Guide & Test Prep
- Required Assignments for Business 302
- U.S. Colonial Resistance Lesson Plans
- Common Core Lesson Plan Templates & Resources
- Team Building Lesson Plans & Resources
- Settings in Literature Lesson Plans & Resources
- California Code of Regulations for Schools
- WV Next Generation Standards for Math
- Continuing Education Opportunities for Microbiology Technologists
- Professional Publications in Literacy
- Dyslexia Programs in Texas
- Study.com's Teacher Edition
- Study.com School Plans

- What is a Test Tube Used for in Science?
- What is Journalism? - Definition, Roles & Issues
- Analogies Lesson for Kids: Definition & Examples
- What is an Interior Designer?
- Atomic Absorptions Spectroscopy: Definition & Uses
- Language Acquisition Lesson Plan
- A Moveable Feast Discussion Questions
- Response to Intervention Assessment Tools
- Quiz & Worksheet - Opposite Numbers
- Quiz & Worksheet - What is Tourism?
- Quiz & Worksheet - Adventure of the Dying Detective Plot Synopsis
- Quiz & Worksheet - What is Apiculture?
- Quiz & Worksheet - How Verbal Learning Works
- Graphs & Charts in Everyday Life Flashcards
- Interpreting & Analyzing Data Sets Flashcards

- HiSET Language Arts - Writing: Prep and Practice
- Holt Science Spectrum - Physical Science with Earth and Space Science: Online Textbook Help
- ORELA English Language Arts: Practice & Study Guide
- UExcel Foundations of Gerontology: Study Guide & Test Prep
- Twelfth Night Study Guide
- ACT Writing - Advanced Skills: Tutoring Solution
- AEPA Middle Grades Math: Basic Algebraic Expressions
- Quiz & Worksheet - Exponential Growth
- Quiz & Worksheet - Powers & Role of Legislatures
- Quiz & Worksheet - Decision Making Models
- Quiz & Worksheet - Characteristics of Market & State-Controlled Economies
- Quiz & Worksheet - Purchases Journal

- Vertebrates: Definition, Features & Classification
- Scientific Approaches in Life, Earth & Physical Science
- Renaissance Lesson Plan
- Script Writing Prompts
- Poetry Writing Prompts
- Inertia Experiments for Kids
- 4th of July Activities for Kids
- Bill of Rights Lesson Plan
- Persuasive Writing Activities
- Multiplication Games for Kids
- Solar System Project Ideas
- Measurement Games for Kids

Browse by subject