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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 45a
Chapter 3, Problem 45a

Find ƒ+g and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

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Step 1: Understand the problem. You are tasked with finding the sum of two functions, f(x) and g(x), which is denoted as (f + g)(x). This means you need to add the two given functions: f(x) = 8x / (x - 2) and g(x) = 6 / (x + 3).
Step 2: Write the expression for (f + g)(x). This is done by adding the two functions: (f + g)(x) = (8x / (x - 2)) + (6 / (x + 3)).
Step 3: Find a common denominator for the two fractions. The denominators are (x - 2) and (x + 3). The least common denominator (LCD) is the product of these two terms: (x - 2)(x + 3).
Step 4: Rewrite each fraction with the common denominator. Multiply the numerator and denominator of the first fraction by (x + 3), and the numerator and denominator of the second fraction by (x - 2). This gives: (f + g)(x) = [(8x)(x + 3) / ((x - 2)(x + 3))] + [(6)(x - 2) / ((x - 2)(x + 3))].
Step 5: Combine the fractions into a single fraction. Add the numerators together while keeping the common denominator: (f + g)(x) = [(8x)(x + 3) + (6)(x - 2)] / ((x - 2)(x + 3)). Simplify the numerator by expanding and combining like terms. Finally, determine the domain by excluding any x-values that make the denominator zero, which are x = 2 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 Addition

Function addition involves combining two functions, f(x) and g(x), to create a new function, denoted as (f + g)(x). This is done by adding the outputs of the two functions for the same input x, resulting in (f + g)(x) = f(x) + g(x). Understanding this concept is crucial for solving the problem as it requires calculating the sum of the given functions.
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Adding & Subtracting Functions Example 1

Domain of a Function

The domain of a function is the set of all possible input values (x) for which the function is defined. For rational functions like f(x) and g(x), the domain is restricted by values that make the denominator zero. Identifying these restrictions is essential to determine the valid inputs for the combined function (f + g).
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Domain Restrictions of Composed Functions

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In this case, f(x) = 8x/(x - 2) and g(x) = 6/(x + 3) are both rational functions. Understanding their behavior, particularly how to handle discontinuities and asymptotes, is important for accurately finding the domain and performing operations like addition.
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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) = x³ − 1

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Textbook Question

Find fg and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

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Textbook Question

Graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = |x|, g(x) = |x| − 2

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Textbook Question

After a 30% price reduction, you purchase a 50″ 4K UHD TV for \$245. What was the television's price before the reduction?

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Textbook Question

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

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Textbook Question

Find f−g and determine the domain for each function. f(x)= = 8x/(x - 2), g(x) = 6/(x+3)

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