In Exercises 31–50, find f/g and determine the domain for each function. f(x) = √(x +4), g(x) = √(x − 1)

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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 written as (f/g)(x) = f(x)/g(x). Additionally, you need to determine the domain of this quotient function.

Step 2: Write the quotient function. Substitute the given functions into the formula: (f/g)(x) = (√(x + 4)) / (√(x − 1)).

Step 3: Determine the domain of f(x). Since f(x) = √(x + 4), the expression inside the square root, x + 4, must be greater than or equal to 0. Solve the inequality x + 4 ≥ 0 to find x ≥ -4. This means the domain of f(x) is all x such that x ≥ -4.

Step 4: Determine the domain of g(x). Since g(x) = √(x − 1), the expression inside the square root, x − 1, must also be greater than or equal to 0. Solve the inequality x − 1 ≥ 0 to find x ≥ 1. This means the domain of g(x) is all x such that x ≥ 1.

Step 5: Combine the domain restrictions. For the quotient (f/g)(x) to be defined, g(x) cannot be 0 (division by zero is undefined). Therefore, x − 1 > 0, which simplifies to x > 1. Combine this with the domain restrictions from f(x) and g(x). The final domain is all x such that x > 1.

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주요 개념

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

Function Division

Function division involves creating a new function by dividing one function by another. In this case, f/g means we take the function f(x) and divide it by g(x). This operation requires understanding how to manipulate functions and the implications of division, particularly regarding the values that make the denominator zero.

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 square root functions, the expression inside the square root must be non-negative. Therefore, determining the domain involves solving inequalities to find the valid x-values for both f(x) and g(x).

Composite Functions

Composite functions occur when one function is applied to the result of another function. In the context of f/g, we need to consider how the outputs of f and g interact, particularly focusing on the restrictions imposed by g(x) since division by zero is undefined. Understanding how to combine and analyze these functions is crucial for finding the overall domain.