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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 45b
Chapter 3, Problem 45b

Find f−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 difference of two functions, f(x) and g(x), which is expressed as (f - g)(x) = f(x) - g(x). Additionally, you need to determine the domain of the resulting function. The given functions are f(x) = 8x / (x - 2) and g(x) = 6 / (x + 3).
Step 2: Write the expression for (f - g)(x). Subtract g(x) from f(x): (f - g)(x) = (8x / (x - 2)) - (6 / (x + 3)). To subtract these fractions, you need a common denominator.
Step 3: Find the common denominator. The denominators of the two fractions are (x - 2) and (x + 3). The least common denominator (LCD) is the product of these two denominators: (x - 2)(x + 3). Rewrite each fraction with this common denominator.
Step 4: Rewrite the fractions with the common denominator. For the first fraction, multiply both numerator and denominator by (x + 3): (8x / (x - 2)) becomes (8x(x + 3) / ((x - 2)(x + 3))). For the second fraction, multiply both numerator and denominator by (x - 2): (6 / (x + 3)) becomes (6(x - 2) / ((x - 2)(x + 3))).
Step 5: Combine the fractions. Now that both fractions have the same denominator, subtract the numerators: (f - g)(x) = [(8x(x + 3)) - (6(x - 2))] / [(x - 2)(x + 3)]. Simplify the numerator by distributing and combining like terms. Finally, determine the domain by identifying the x-values that make the denominator zero. These are x = 2 and x = -3, which must be excluded from the domain.

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Key Concepts

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

Function Subtraction

Function subtraction involves finding the difference between two functions, denoted as (f - g)(x) = f(x) - g(x). In this case, you will subtract the output of g(x) from f(x) for each value of x in the domain where both functions are defined.
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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 like f(x) and g(x), the domain excludes values that make the denominator zero, as these would result in undefined outputs.
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Finding Common Denominators

When subtracting two rational functions, it is often necessary to find a common denominator to combine the fractions. This involves identifying the least common multiple of the denominators, which allows for the proper subtraction of the two functions while maintaining equivalent values.
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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

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