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 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). Additionally, you need to determine the domain of the resulting function.

Step 2: Write the expression for (f−g)(x). Subtract g(x) from f(x): (f−g)(x) = f(x) − g(x). Substituting the given functions, this becomes (f−g)(x) = √(x + 4) − √(x − 1).

Step 3: Analyze the domain of f(x). The square root function √(x + 4) is defined only when the expression inside the square root is non-negative. Solve x + 4 ≥ 0 to find the domain restriction for f(x).

Step 4: Analyze the domain of g(x). Similarly, the square root function √(x − 1) is defined only when the expression inside the square root is non-negative. Solve x − 1 ≥ 0 to find the domain restriction for g(x).

Step 5: Combine the domain restrictions. The domain of (f−g)(x) is the intersection of the domains of f(x) and g(x). This means you need to find the values of x that satisfy both domain conditions simultaneously.

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주요 개념

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

Function Operations

Function operations involve combining two or more functions to create new functions. In this case, f(x) and g(x) are being combined through subtraction (f - g). Understanding how to perform operations on functions is essential for finding the resulting function and analyzing its properties.

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 functions involving square roots, the expression inside the root must be non-negative. Therefore, determining the domain requires identifying the values of x that satisfy these conditions for both f(x) and g(x).

Combining Domains

When subtracting two functions, the domain of the resulting function (f - g) is determined by the intersection of the individual domains of f(x) and g(x). This means that the values of x must be valid for both functions simultaneously, ensuring that the resulting function is defined across the entire domain.