Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 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)

Verified step by step guidance

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.

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.

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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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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