Find f−g 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 difference of two functions, f(x) and g(x), denoted as (f - g)(x). This means you need to subtract g(x) from f(x). The functions are f(x) = (5x + 1) / (x² - 9) and g(x) = (4x - 2) / (x² - 9).

Step 2: Write the expression for (f - g)(x). Subtract g(x) from f(x): (f - g)(x) = f(x) - g(x) = [(5x + 1) / (x² - 9)] - [(4x - 2) / (x² - 9)].

Step 3: Combine the fractions. Since the denominators are the same (x² - 9), you can combine the numerators directly: (f - g)(x) = [(5x + 1) - (4x - 2)] / (x² - 9).

Step 4: Simplify the numerator. Distribute the negative sign in the second term: (5x + 1) - (4x - 2) = 5x + 1 - 4x + 2 = (5x - 4x) + (1 + 2) = x + 3. So, (f - g)(x) = (x + 3) / (x² - 9).

Step 5: Determine the domain. The domain of a function is the set of all x-values for which the function is defined. The denominator x² - 9 cannot be zero, as division by zero is undefined. Solve x² - 9 = 0 to find the restricted values: x² = 9, so x = ±3. Therefore, the domain is all real numbers except x = 3 and x = -3.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Function Operations

Function operations involve combining two functions to create a new function. In this case, f-g means subtracting the function g(x) from f(x). Understanding how to perform operations on functions is essential for manipulating and analyzing them effectively.

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, the domain is restricted by values that make the denominator zero. Identifying the domain is crucial for ensuring that the function behaves correctly and does not produce undefined values.

Rational Functions

Rational functions are ratios of two polynomials. They can exhibit unique behaviors, such as asymptotes and discontinuities, particularly where the denominator equals zero. Understanding the properties of rational functions helps in analyzing their graphs and determining their domains.