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Composition of Functions

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Composition of Functions

Study Guide

Key Definition

The function whose value at $x$ is $f(g(x))$ is called the composite of the functions $f$ and $g$.

Important Notes

The operation that combines $f$ and $g$ to produce the composite is called composition.
Notation for the composite function can be written as $f ∘ g$ or $f(g(x))$.
The domain of $f(g(x))$ is the set of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.
Example: If $f(x) = x$ and $g(x) = 3x - 5$, then find $f(g(4))$.
Solution Example: Substituting 4 for $x$ in the definition of the function $g$, $g(4) = 3(4) - 5 = 7$. Then, $f(g(4)) = f(7)$, so $f(7) = 7$.

Mathematical Notation

$f ∘ g$ or $f(g(x))$ notation represents the composition of functions $f$ and $g$.

Remember to use proper notation when solving problems

Why It Works

Composition of functions works because it allows us to evaluate a function at the output of another function. It is a common technique used in mathematics to create new functions from existing ones.

Remember

Always remember that for the composite function $f(g(x))$, x must be in the domain of $g$ and $g(x)$ must be in the domain of $f$.

Quick Reference

Composition of Functions:$f(g(x))$

Understanding Composition of Functions

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In the composition of functions, one function is applied first and then the second function is applied to the result of the first function.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $f(x) = x^2$ and $g(x) = 2x - 3$, what is the value of $f(g(2))$?

Real-World Problem

Question Exercise

Intermediate

Music Download Scenario

You pay a monthly subscription fee and an additional fee for each song you download. The per-song fee is represented by $g(x) = 0.99x$. To model the total cost, define $f(x) = x + 10$ (where $x$ is the download cost). Write a composite function to represent the total cost if you download $x$ songs.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given the functions $f(x) = 2x + 1$ and $g(x) = 3x - 2$, find the composite function $f(g(x))$ and simplify.

Challenge Quiz

Single Choice Quiz

Advanced

Given the functions $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, what is the value of $g(f(2))$?

Recap

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