Find fg and determine the domain for each function. f(x)= = (5x+1)/(x² - 9), g(x) = (4x -2)/(x² - 9)

검증된 단계별 안내

Step 1: Understand the problem. You are tasked with finding the composition of two functions, fg(x), which means f(g(x)). Additionally, you need to determine the domain of the resulting function. The given functions are f(x) = (5x + 1) / (x² - 9) and g(x) = (4x - 2) / (x² - 9).

Step 2: Substitute g(x) into f(x) to find fg(x). Replace every instance of x in f(x) with g(x). This gives fg(x) = (5 * [(4x - 2) / (x² - 9)] + 1) / ([(4x - 2) / (x² - 9)]² - 9).

Step 3: Simplify the numerator of fg(x). Distribute the 5 to the terms in g(x), then add 1. This results in the numerator: (5 * (4x - 2) / (x² - 9)) + 1 = (20x - 10) / (x² - 9) + 1. Combine terms over a common denominator.

Step 4: Simplify the denominator of fg(x). Square g(x) to get [(4x - 2) / (x² - 9)]², then subtract 9. This results in: [(4x - 2)² / (x² - 9)²] - 9. Combine terms over a common denominator.

Step 5: Determine the domain of fg(x). The domain is restricted by values of x that make the denominator of f(x), g(x), or fg(x) equal to zero. For both f(x) and g(x), the denominator x² - 9 = 0 when x = ±3. Additionally, check for any restrictions introduced by fg(x). Exclude these values from the domain.

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주요 개념

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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.

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 valid inputs for the composite function fg.

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 the properties of rational functions is key to analyzing their domains and compositions.