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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 2 - Functions and GraphsProblem 43b
Chapter 3, Problem 43b

Find f−g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

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Step 1: Understand the problem. You are tasked with finding the difference of two functions, f(x) and g(x), denoted as (f - g)(x). This means you need to subtract g(x) from f(x). The functions are f(x) = (5x + 1) / (x² - 9) and g(x) = (4x - 2) / (x² - 9).
Step 2: Write the expression for (f - g)(x). Subtract g(x) from f(x): (f - g)(x) = f(x) - g(x) = [(5x + 1) / (x² - 9)] - [(4x - 2) / (x² - 9)].
Step 3: Combine the fractions. Since the denominators are the same (x² - 9), you can combine the numerators directly: (f - g)(x) = [(5x + 1) - (4x - 2)] / (x² - 9).
Step 4: Simplify the numerator. Distribute the negative sign in the second term: (5x + 1) - (4x - 2) = 5x + 1 - 4x + 2 = (5x - 4x) + (1 + 2) = x + 3. So, (f - g)(x) = (x + 3) / (x² - 9).
Step 5: Determine the domain. The domain of a function is the set of all x-values for which the function is defined. The denominator x² - 9 cannot be zero, as division by zero is undefined. Solve x² - 9 = 0 to find the restricted values: x² = 9, so x = ±3. Therefore, the domain is all real numbers except x = 3 and x = -3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Function Operations

Function operations involve combining two functions to create a new function. In this case, f-g means subtracting the function g(x) from f(x). Understanding how to perform operations on functions is essential for manipulating and analyzing them effectively.
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Multiplying & Dividing Functions

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by values that make the denominator zero. Identifying the domain is crucial for ensuring that the function behaves correctly and does not produce undefined values.
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Domain Restrictions of Composed Functions

Rational Functions

Rational functions are ratios of two polynomials. They can exhibit unique behaviors, such as asymptotes and discontinuities, particularly where the denominator equals zero. Understanding the properties of rational functions helps in analyzing their graphs and determining their domains.
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Related Practice
Textbook Question

In Exercises 39-52, a. Find an equation for ƒ¯¹(x). b. Graph ƒ and ƒ¯¹(x) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range off and ƒ¯¹. f(x) = (x − 1)², x ≤ 1

Textbook Question

Use the graph of y = f(x) to graph each function g. g(x) = (1/2)f(2x)

Textbook Question

Find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x)= = (3x+1)/(x² - 25), g(x) = (2x -4)/(x² - 25)

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Textbook Question

Find f/g and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

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Textbook Question

Find ƒ+g and determine the domain for each function.

f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

Textbook Question

Find fg and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

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