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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 2 - Functions and Graphs문제 45b
3장, 문제 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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
5:56
Adding & Subtracting 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 like f(x) and g(x), the domain excludes values that make the denominator zero, as these would result in undefined outputs.
추천 영상:
3:51
Domain Restrictions of Composed Functions

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.
추천 영상:
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Rationalizing Denominators
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