Subjects

Algebra 2

Modeling

Modeling with Function Combination

Algebra 2

Dec 1, 2025

3 pages

Mastering Composite Functions and Operations

sumehra @sumehra

Combining functions allows us to create more complex mathematical relationships from simpler ones. You'll learn how to add,... Show more

2.7 Combining Functions

I. Sums, Differences, Products, and Quotients

Given two functions f and g, we define the new function f+g by (f+g)

Combining Functions Arithmetic Operations

Ever wondered how mathematicians build complex functions from simpler ones? It's like creating a recipe by combining ingredients! When we have two functions f and g, we can combine them in several ways.

The four basic operations create new functions with specific domains

Sum $f+g$ (x) = f(x) + g(x)
Difference $f-g$ (x) = f(x) - g(x)
Product (fg)(x) = f(x) × g(x)
Quotient $f/g$ (x) = f(x) ÷ g(x), where g(x) ≠ 0

For example, if f(x) = 3x+1 and g(x) = 2x²-1, we can find

$g-f$ (x) = 2x² - $3x+1$ = 2x² - 3x - 2
$f+g$ (x) = $3x+1$ + $2x²-1$ = 2x² + 3x
$g/f$ (x) = $2x²-1$ ÷ $3x+1$ , which is defined when x ≠ -⅓

Pro Tip When finding domains of combined functions, remember that both functions need to be defined at that input value. For quotients, you also need to check where the denominator equals zero!

Function Composition

Function composition is like a mathematical assembly line! When we write (f ∘ g)(x) = f(g(x)), we're saying "take input x, run it through function g, then use that result as input for function f."

The domain of f ∘ g includes all values of x where

x is in the domain of g
g(x) is in the domain of f

Let's practice with f(x) = 3x+1 and g(x) = 2x²-1

(f ∘ g)(x) = f(g(x)) = f $2x²-1$ = 3 $2x²-1$ +1 = 6x²-2
(g ∘ f)(x) = g(f(x)) = g $3x+1$ = 2 $3x+1$ ²-1 = 18x²+12x+1

Notice that (f ∘ g)(x) ≠ (g ∘ f)(x) in most cases. Order matters in function composition!

Remember Function composition isn't commutative! Think of it like putting on socks and shoes—you can't switch the order and get the same result.

Advanced Composition and Decomposition

You can compose more than just two functions! For three functions, we follow the pattern (f ∘ g ∘ h)(x) = f(g(h(x))). This means we apply h first, then g, and finally f.

For example, if f(x) = x/ $x+1$ , g(x) = x¹⁰, and h(x) = x+3, then

(f ∘ g ∘ h)(x) = f(g(h(x))) = f $g(x+3)$ = f $(x+3)¹⁰$ = $x+3$ ¹⁰/ $(x+3)¹⁰+1$

Function decomposition is like working backward—finding simpler functions that compose to make a more complex one. For F(x) = √ $x+9$ , we can decompose it into

g(x) = x+9 (the inner function)
f(x) = √x (the outer function)
(f ∘ g)(x) = f(g(x)) = f $x+9$ = √ $x+9$ = F(x)

Challenge yourself When you see a complex function, try to identify if it could be written as a composition of simpler functions. This skill helps solve many calculus problems later!

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Subjects

Algebra 2

Modeling

Modeling with Function Combination

Algebra 2

•

Dec 1, 2025

•

3 pages

Mastering Composite Functions and Operations

sumehra

@sumehra

Combining functions allows us to create more complex mathematical relationships from simpler ones. You'll learn how to add, subtract, multiply, and compose functions—skills that are essential for modeling real-world situations and solving advanced math problems.

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Combining Functions: Arithmetic Operations

The four basic operations create new functions with specific domains:

Sum: $f+g$ (x) = f(x) + g(x)
Difference: $f-g$ (x) = f(x) - g(x)
Product: (fg)(x) = f(x) × g(x)
Quotient: $f/g$ (x) = f(x) ÷ g(x), where g(x) ≠ 0

For example, if f(x) = 3x+1 and g(x) = 2x²-1, we can find:

$g-f$ (x) = 2x² - $3x+1$ = 2x² - 3x - 2
$f+g$ (x) = $3x+1$ + $2x²-1$ = 2x² + 3x
$g/f$ (x) = $2x²-1$ ÷ $3x+1$ , which is defined when x ≠ -⅓

Pro Tip: When finding domains of combined functions, remember that both functions need to be defined at that input value. For quotients, you also need to check where the denominator equals zero!

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Function Composition

Function composition is like a mathematical assembly line! When we write (f ∘ g)(x) = f(g(x)), we're saying "take input x, run it through function g, then use that result as input for function f."

The domain of f ∘ g includes all values of x where:

x is in the domain of g
g(x) is in the domain of f

Let's practice with f(x) = 3x+1 and g(x) = 2x²-1:

(f ∘ g)(x) = f(g(x)) = f $2x²-1$ = 3 $2x²-1$ +1 = 6x²-2
(g ∘ f)(x) = g(f(x)) = g $3x+1$ = 2 $3x+1$ ²-1 = 18x²+12x+1

Notice that (f ∘ g)(x) ≠ (g ∘ f)(x) in most cases. Order matters in function composition!

Remember: Function composition isn't commutative! Think of it like putting on socks and shoes—you can't switch the order and get the same result.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Advanced Composition and Decomposition

You can compose more than just two functions! For three functions, we follow the pattern (f ∘ g ∘ h)(x) = f(g(h(x))). This means we apply h first, then g, and finally f.

For example, if f(x) = x/ $x+1$ , g(x) = x¹⁰, and h(x) = x+3, then:

(f ∘ g ∘ h)(x) = f(g(h(x))) = f $g(x+3)$ = f $(x+3)¹⁰$ = $x+3$ ¹⁰/ $(x+3)¹⁰+1$

Function decomposition is like working backward—finding simpler functions that compose to make a more complex one. For F(x) = √ $x+9$ , we can decompose it into:

g(x) = x+9 (the inner function)
f(x) = √x (the outer function)
(f ∘ g)(x) = f(g(x)) = f $x+9$ = √ $x+9$ = F(x)

Challenge yourself: When you see a complex function, try to identify if it could be written as a composition of simpler functions. This skill helps solve many calculus problems later!

We thought you’d never ask...

What is the Knowunity AI companion?

Where can I download the Knowunity app?

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Is Knowunity really free of charge?

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4.9/5

App Store

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Stefan S

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Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

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Greenlight Bonnie

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Elisha

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Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user