Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321979476

10th Edition

5,609 Questions

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320,854 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

Composite functions enable us to combine two function operations into a single process, where the output of the inner function becomes the input of the outer function. It is crucial to assess the domain of each component to ensure the resulting function is well-defined. Examples such as calculating the area of an oil slick illustrate real-world applications of composite functions and the importance of function decomposition.

Learning Objectives

1

Form and evaluate composite functions from given component functions.

2

Determine the domain of composite functions by analyzing the domains of the individual functions.

3

Apply composite functions to real-world scenarios, such as modeling the area of an oil slick.

4

Decompose a composite function into its constituent functions and understand the order of operation.

Key Concepts

CONCEPT

DEFINITION

Composite Function

A function defined by applying one function to the results of another, denoted by (f ∘ g)(x) = f(g(x)).

Domain of a Composite Function

The set of all x-values in the domain of g such that g(x) is in the domain of f, ensuring that the composite function f(g(x)) is defined.

Decomposition

The process of breaking a composite function into its component functions, typically by letting u = g(x) and then expressing f(u).

Example Problems

Example 1

Find $f(3)$ if $f(x)=-4 x^{2}+5 x$

Example 2

Find $f(3 x)$ if $f(x)=4-2 x^{2}$

Example 3

Find the domain of the function $f(x)=\frac{x^{2}-1}{x^{2}-25}$

Example 4

Given two functions $f$ and $g,$ the _________ _________, denoted $f \circ g,$ is defined by $(f \circ g)(x)=$ ________.

Example 5

True or False If $f(x)=x^{2}$ and $g(x)=\sqrt{x+9},$ then $(f \circ g)(4)=5 .$
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Step-by-Step Explanations

QUESTION

Given f(x) = 2x² - 3 and g(x) = 4x, find (f ∘ g)(1).

STEP-BY-STEP ANSWER:

Step 1: Substitute x into g: g(1) = 4(1) = 4.
Step 2: Substitute g(1) into f: f(4) = 2(4)² - 3.
Step 3: Compute (4)² = 16, then multiply: 2 × 16 = 32.
Step 4: Subtract 3: 32 - 3 = 29.
Final Answer: (f ∘ g)(1) = 29.

Forming a Composite Function

QUESTION

If f(x) = 1/(x+2) and g(x) = 4/(x-1), find the domain of (f ∘ g)(x).

STEP-BY-STEP ANSWER:

Step 1: Determine the domain of g(x): g is undefined when x - 1 = 0, so x ≠ 1.
Step 2: Find g(x) and identify when f is undefined: f(u) = 1/(u+2) is undefined when u + 2 = 0, i.e., when u = -2.
Step 3: Set g(x) ≠ -2. Solve 4/(x-1) = -2: Multiply both sides by (x-1) → 4 = -2(x-1).
Step 4: Solve: 4 = -2x + 2, hence 2x = -2, so x = -1.
Step 5: Combine restrictions: x ≠ 1 and x ≠ -1.
Final Answer: The domain of (f ∘ g)(x) is all real numbers except x = 1 and x = -1.

Finding the Domain of a Composite Function

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Common Mistakes

  • Failing to check that the inner function’s output falls within the domain of the outer function.
  • Reversing the order of functions, leading to incorrect evaluations of f ? g versus g ? f.
  • Not properly simplifying the composite function expression after substitution.
  • Overlooking restrictions in the domain inherent to each individual function.