Find fg and determine the domain for each function. f(x) = √x, g(x) = x − 4

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Step 1: Understand the problem. The goal is to find the composition of the functions f and g, denoted as fg(x) = f(g(x)). This means we will substitute g(x) into f(x).

Step 2: Substitute g(x) = x − 4 into f(x) = √x. This gives fg(x) = √(x − 4).

Step 3: Analyze the domain of fg(x). The square root function √x is only defined for x ≥ 0. Therefore, for fg(x) = √(x − 4), the expression inside the square root, x − 4, must be greater than or equal to 0.

Step 4: Solve the inequality x − 4 ≥ 0 to determine the domain. Add 4 to both sides to get x ≥ 4.

Step 5: Conclude that the domain of fg(x) is all x such that x ≥ 4. In interval notation, the domain is [4, ∞).

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주요 개념

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

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 the function f(x) = √x, the domain is restricted to non-negative values of x, as square roots of negative numbers are not defined in the real number system. Identifying the domain of both f and g is essential for determining the overall domain of the composite function.

Square Root Function

The square root function, denoted as f(x) = √x, outputs the non-negative square root of x. This function is only defined for x ≥ 0, which directly impacts the domain when composing it with another function. Understanding the properties of square root functions is vital for correctly evaluating and determining the domain of the composite function fg.