2.7 Composition Of Functions Practice Set 1

2.7 Composition of Functions: Practice Set 1 – A Deep Dive

This comprehensive guide delves into the intricacies of composition of functions, specifically focusing on Practice Set 1 (assuming a standard curriculum). We will not only solve problems but also explore the underlying concepts, providing you with a robust understanding to tackle any composition of function challenge. This guide is designed to be highly searchable and informative, incorporating SEO best practices for optimal online visibility.

Understanding Composition of Functions

Before we jump into Practice Set 1, let's solidify our understanding of function composition. Function composition is a mathematical operation that combines two or more functions to create a new function. Instead of applying functions sequentially, composition creates a single function that incorporates the operations of both.

The notation for composition is crucial. We typically write the composition of functions f and g as (f ∘ g)(x), which is read as "f composed with g of x." This means we first apply function g to x, and then apply function f to the result.

Key Concept: The output of the inner function (g(x)) becomes the input of the outer function (f(x)).

Visualizing Composition

Think of functions as machines. Each machine takes an input, processes it, and produces an output. Composition is like connecting these machines in a series. The output of the first machine becomes the input of the second.

Example:

Let's say we have:

• f(x) = x²
• g(x) = x + 2

Then, (f ∘ g)(x) = f(g(x)) = f(x + 2) = (x + 2)²

This means we first add 2 to x (using g(x)), and then square the result (using f(x)).

Practice Set 1: Problem Solving and Explanations

Now, let's tackle a hypothetical Practice Set 1. Remember that the specific problems will vary depending on your textbook or course materials. The examples below represent a typical range of difficulty and cover key concepts. We'll focus on detailed explanations to enhance your understanding.

Problem 1:

Given f(x) = 3x - 1 and g(x) = x² + 2, find (f ∘ g)(x) and (g ∘ f)(x).

Solution:

(f ∘ g)(x) = f(g(x)) = f(x² + 2) = 3(x² + 2) - 1 = 3x² + 6 - 1 = 3x² + 5

We substitute g(x) into f(x).

(g ∘ f)(x) = g(f(x)) = g(3x - 1) = (3x - 1)² + 2 = 9x² - 6x + 1 + 2 = 9x² - 6x + 3

We substitute f(x) into g(x). Notice that (f ∘ g)(x) ≠ (g ∘ f)(x), highlighting that function composition is not commutative.

Problem 2:

Find the domain of (f ∘ g)(x) given f(x) = √x and g(x) = x - 4.

Solution:

First, let's find (f ∘ g)(x):

(f ∘ g)(x) = f(g(x)) = f(x - 4) = √(x - 4)

The domain of a square root function is restricted to non-negative values inside the square root. Therefore, we need x - 4 ≥ 0, which implies x ≥ 4.

The domain of (f ∘ g)(x) is [4, ∞).

Problem 3:

If h(x) = (2x + 1)³, find functions f(x) and g(x) such that h(x) = (f ∘ g)(x).

Solution:

This problem tests your ability to decompose a function. We need to find two simpler functions whose composition results in h(x). There are multiple possible solutions, but one straightforward approach is:

Let g(x) = 2x + 1 and f(x) = x³.

Then, (f ∘ g)(x) = f(g(x)) = f(2x + 1) = (2x + 1)³, which is equal to h(x).

Problem 4:

Given f(x) = 1/x and g(x) = x + 3, find (f ∘ g)(2) and (g ∘ f)(2).

Solution:

(f ∘ g)(2) = f(g(2)) = f(2 + 3) = f(5) = 1/5

(g ∘ f)(2) = g(f(2)) = g(1/2) = (1/2) + 3 = 7/2

Problem 5 (Challenge Problem):

Let f(x) = ax + b and g(x) = cx + d. Find (f ∘ g)(x) and (g ∘ f)(x). Under what conditions are (f ∘ g)(x) and (g ∘ f)(x) equal?

Solution:

(f ∘ g)(x) = f(g(x)) = f(cx + d) = a(cx + d) + b = acx + ad + b

(g ∘ f)(x) = g(f(x)) = g(ax + b) = c(ax + b) + d = acx + bc + d

For (f ∘ g)(x) and (g ∘ f)(x) to be equal, their coefficients must be equal:

• ad + b = bc + d

This equation reveals the conditions under which the composition of these linear functions is commutative. It's important to note this is not generally true for all functions.

Expanding Your Understanding: Beyond Practice Set 1

This Practice Set 1 serves as a foundation. To master composition of functions, consider exploring these advanced concepts:

• Composition of more than two functions: Extend the concept to three or more functions, understanding the order of operations is critical.
• Composition with piecewise functions: Practice composing functions where individual functions are defined over different intervals.
• Inverse functions and composition: Understand how composition interacts with inverse functions; (f ∘ f⁻¹)(x) = x, provided the inverse exists.
• Applications of composition: Explore real-world applications in areas like physics, engineering, and computer science.

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