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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 45c
3장, 문제 45c

Find fg 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 composition of two functions, fg(x), which means f(g(x)). This involves substituting g(x) into f(x). Additionally, you need to determine the domain of the resulting function, which is the set of all x-values for which the function is defined.
Step 2: Write the composition f(g(x)). Start by substituting g(x) = 6/(x + 3) into f(x) = 8x / (x - 2). This gives f(g(x)) = 8 * (6 / (x + 3)) / ((6 / (x + 3)) - 2).
Step 3: Simplify the numerator and denominator. The numerator becomes 8 * (6 / (x + 3)) = 48 / (x + 3). The denominator is (6 / (x + 3)) - 2. To simplify the denominator, rewrite 2 as (2(x + 3)) / (x + 3), so the denominator becomes (6 / (x + 3)) - (2(x + 3) / (x + 3)). Combine the terms in the denominator under a common denominator.
Step 4: Combine the terms in the denominator. The denominator becomes (6 - 2(x + 3)) / (x + 3). Simplify the numerator of the denominator: 6 - 2(x + 3) = 6 - 2x - 6 = -2x. So the denominator becomes -2x / (x + 3).
Step 5: Write the final simplified expression for f(g(x)). Substitute the simplified numerator and denominator into the composition: f(g(x)) = (48 / (x + 3)) / (-2x / (x + 3)). Simplify by multiplying by the reciprocal of the denominator: f(g(x)) = (48 / (x + 3)) * ((x + 3) / -2x). Cancel out (x + 3) (as long as x ≠ -3), leaving f(g(x)) = 48 / (-2x). Finally, determine the domain by excluding values that make any denominator zero: x ≠ -3 (from g(x)) and x ≠ 0 (from the final expression).

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주요 개념

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

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, fg means f(g(x)), which requires substituting g(x) into f(x). Understanding how to perform this substitution is crucial for finding the composite function.
추천 영상:
4:56
Function Composition

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 is restricted by values that make the denominator zero. Identifying these restrictions is essential for determining the overall domain of the composite function.
추천 영상:
3:51
Domain Restrictions of Composed Functions

Rational Functions

Rational functions are ratios of polynomials, expressed in the form f(x) = P(x)/Q(x), where P and Q are polynomials. The behavior of these functions, including their asymptotes and discontinuities, is influenced by the zeros of the denominator. Understanding rational functions is key to analyzing their domains and compositions.
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Intro to Rational Functions
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