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

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Step 1: Understand the problem. You are tasked with finding the quotient of two functions, f(x) and g(x), which is represented as (f/g)(x) = f(x)/g(x). Additionally, you need to determine the domain of the resulting function.

Step 2: Write the given functions. f(x) = (5x + 1)/(x² - 9) and g(x) = (4x - 2)/(x² - 9). To find (f/g)(x), divide f(x) by g(x): (f/g)(x) = [f(x)] / [g(x)].

Step 3: Simplify the division. Dividing two fractions involves multiplying the numerator of the first fraction by the reciprocal of the second fraction. So, (f/g)(x) = [(5x + 1)/(x² - 9)] * [(x² - 9)/(4x - 2)].

Step 4: Cancel out common factors. Notice that (x² - 9) appears in both the numerator and denominator, so it cancels out, leaving (f/g)(x) = (5x + 1)/(4x - 2).

Step 5: Determine the domain. The domain of a function is the set of all x-values for which the function is defined. For (f/g)(x), the denominator of both f(x) and g(x) must not be zero, and g(x) itself must not be zero. Solve x² - 9 = 0 to find x-values that make the denominator zero, and solve 4x - 2 = 0 to find x-values that make g(x) zero. Exclude these values from the domain.

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주요 개념

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Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. In this case, f(x) and g(x) are both rational functions, where the numerator and denominator are polynomials. Understanding how to manipulate and simplify these functions is crucial for finding their quotient and determining their domains.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is restricted by values that make the denominator zero, as division by zero is undefined. Identifying these restrictions is essential for determining the valid inputs for f/g.

Quotient of Functions

The quotient of two functions, denoted as f/g, is calculated by dividing the output of function f by the output of function g. This operation requires careful attention to the domains of both functions to ensure that the resulting function is defined. Additionally, simplifying the quotient may involve factoring and canceling common terms in the numerator and denominator.