#🚀 AP Pre-Calculus: Composition of Functions - Your Night-Before-Exam Guide! 🚀

Hey there! Let's make sure you're totally prepped for the exam. This guide is designed to be your quick, high-impact review for composite functions. We'll keep it engaging, clear, and focused on what you really need to know. Let's dive in!

#🧩 What Are Composite Functions?

Composite functions are like a function sandwich 🥪—you take the output of one function and feed it as the input to another. It's all about chaining functions together.

#The Basics

Notation: f(g(x)) means you first apply g to x, and then you apply f to the result. Think of it as working from the inside out.
Key Idea: The output of the inner function becomes the input of the outer function.

Composition of function example displayed: Find f(g(x)), where g(x) equal to x squared (x^2) and f(x) equal to x+3. Answer: f(g(x)) is equal to x squared plus 3 (x^2+3).

Example: If g(x) = x² and f(x) = x + 3, then f(g(x)) = f(x²) = x² + 3

#Graphical Approach

You can also find composite function values using graphs. Find the output of g(x) from its graph, and then use that output as the input for f(x) on its graph. 📊

#📝 Notation and Properties

#Function Notation

f(g(x)) is the same as f ∘ g(x), which means "f of g of x."

Key Concept

Order Matters! Composition is not commutative. Usually, f(g(x)) ≠ g(f(x)). Always be mindful of the order of operations. 💡

#Identity Function

The identity function is f(x) = x. When you compose it with any other function g(x), you get g(x) back. Think of it like adding 0 or multiplying by 1—it doesn't change the original function. 🤯

Memory Aid

Think of the identity function f(x) = x as a mirror: whatever you put in, it reflects back unchanged. This is why g(f(x)) = f(g(x)) = g(x).

#🧑🏿‍💻 Working with Composite Functions

#Analytic Representation

To find f(g(x)) analytically, substitute the entire function g(x) for every x in f(x). This is function composition in action!

#Numerical and Graphical Methods

Numerical: Use a table of values for g(x) as inputs for f(x).
Graphical: Use the graph of g(x) to find outputs, then use those as inputs for the graph of f(x).

Graph f(x) and g(x) plotted on a coordinate plane with a composite function: f•g.

Visualizing f(g(x)) using graphs

#🔨 Function Decomposition

Decomposition: Breaking a complex function into simpler parts. It’s like taking apart a machine to see how it works. 🧸
Substitution Property: Ensure the variable in one function replaces each instance of the function it was composed with.

Common Mistake

When decomposing, make sure you're replacing every instance of the variable. It's easy to miss one!

#🦸🏽 Transformation Time!

Composite functions can also help us understand transformations of functions. Let's explore additive and multiplicative transformations. 💄

#Additive Transformations

Vertical Translation: g(x) = x + k composed with f(x) shifts f(x) up or down by k units. If k is positive, it shifts up; if k is negative, it shifts down.

#Multiplicative Transformations

Horizontal Dilation: g(x) = kx composed with f(x) stretches or shrinks f(x) horizontally by a factor of k. If k > 1, it shrinks; if 0 < k < 1, it stretches. 😁

Exam Tip

Remember: Transformations are all about how the input to f(x) is being changed. Additive changes shift, and multiplicative changes stretch or shrink.

#🎯 Final Exam Focus

High-Value Topics: Function composition, order of operations, and transformations.
Common Question Types: Finding f(g(x)) and g(f(x)) analytically, using graphs, and function decomposition.
Time Management: Practice quick substitutions and graphical analysis.
Common Pitfalls: Forgetting the order of operations and missing variables during substitutions.

#📝 Practice Questions

Practice Question

#Multiple Choice Questions

Given f(x) = 2x - 3 and g(x) = x² + 1, find f(g(2)). (A) 6 (B) 7 (C) 9 (D) 11
If h(x) = √(x) and k(x) = 4x + 5, what is h(k(x))? (A) 2√(x) + √(5) (B) √(4x + 5) (C) 2x + √(5) (D) 4√(x) + 5
The graphs of f(x) and g(x) are shown below. What is the approximate value of f(g(1))?

(A graph would be provided here showing f(x) and g(x) and their values at x=1)

(A) 1 (B) 2 (C) 3 (D) 4

#Free Response Question

Let f(x) = 3x + 2 and g(x) = x² - 4.

(a) Find f(g(x)).

(b) Find g(f(x)).

(d) Find x such that f(g(x)) = 29.

Scoring Breakdown:

(a) 2 points: 1 point for correct substitution, 1 point for correct simplification. f(g(x)) = 3(x² - 4) + 2 = 3x² - 12 + 2 = 3x² - 10

(b) 2 points: 1 point for correct substitution, 1 point for correct simplification. g(f(x)) = (3x + 2)² - 4 = 9x² + 12x + 4 - 4 = 9x² + 12x

(c) 2 points: 1 point for correct substitution, 1 point for correct evaluation. f(g(2)) = 3(2²) - 10 = 3(4) - 10 = 12 - 10 = 2

(d) 3 points: 1 point for setting up the equation, 1 point for algebraic manipulation, 1 point for the correct answer. 3x² - 10 = 29 => 3x² = 39 => x² = 13 => x = ±√13

Remember, you've got this! Focus on understanding the process of function composition, and you'll be well-prepared for anything the AP exam throws your way. Good luck! 🍀