Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321716811

9th Edition

5,201 Questions

Group icon
40,131 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section on composite functions focuses on evaluating functions where one function is applied to the result of another and determining the domain of such composite functions. Students learn to carefully process the 'inside' function before applying the 'outside' function and must consider domain restrictions from both. Numerous examples, including graphical and calculator verifications, demonstrate how composite functions are used both in pure mathematics and in real-world applications.

Learning Objectives

1

Evaluate composite functions by correctly applying the 'inside' and 'outside' function methodology.

2

Determine the domain of composite functions by considering the restrictions on both the inner function and the outer function.

3

Apply techniques using graphing calculators to verify and analyze composite functions graphically.

4

Decompose complex functions into simpler components to understand and solve real-world application problems.

Key Concepts

CONCEPT

DEFINITION

Composite Function

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

Domain

The set of input values for which a function is defined. For composite functions, x must be in the domain of the inner function and f(g(x)) must be defined in the domain of the outer function.

Inside Function

The function that is applied first in a composite function, typically represented by g(x) in f(g(x)).

Outside Function

The function that is applied to the result of the inside function in a composite function, typically represented by f in f(g(x)).

Graphing Calculator

A tool used to evaluate and graph complex functions, including composite functions, to aid in visual understanding and verification of solutions.

Example Problems

Example 1

$\text { Find } f(3) \text { if } f(x)=-4 x^{2}+5 x .(\mathrm{pp} .203-206)$

Example 2

$\text { Find } f(3 x) \text { if } f(x)=4-2 x^{2},(p p, 203-206)$

Example 3

Find the domain of the function $f(x)=\frac{x^{2}-1}{x^{2}-25}$ $(p p .206-208)$

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 $f(g(x))=f(x) \cdot g(x)$.
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

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

STEP-BY-STEP ANSWER:

Step 1: Evaluate the inside function by finding g(1). Since g(x) = 4x, then g(1) = 4*1 = 4.
Step 2: Substitute the result of g(1) into the outside function f. That is, compute f(4).
Step 3: Since f(x) = 2x² - 3, then f(4) = 2*(4)² - 3 = 2*16 - 3 = 32 - 3 = 29.
Final Answer: (f ∘ g)(1) = 29.

Evaluating (f ∘ g)(1)

QUESTION

If f(x) and g(x) both have the domain of all real numbers, what is the domain of (f ∘ g)(x)?

STEP-BY-STEP ANSWER:

Step 1: Identify the domain of the inner function g(x) which is all real numbers.
Step 2: Note that since g(x) produces all real numbers and the outer function f(x) is defined for all real numbers, no additional restrictions occur.
Step 3: Conclude that the domain of the composite function (f ∘ g)(x) is all real numbers.
Final Answer: Domain of (f ∘ g)(x) is all real numbers.

Determining the Domain of a Composite Function

Scroll left
Scroll right

Common Mistakes

  • Confusing the order of composition by applying the outer function first instead of the inner function.
  • Neglecting to adjust the domain to account for restrictions imposed by the inner function's output.
  • Overlooking potential division by zero or undefined expressions when determining the domain.
  • Assuming that the composite function's domain is automatically the same as the domain of the inner function without verifying the outer function's constraints.