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

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Step 1: Understand the problem. The goal is to find the composition of the two functions, fg(x), which means f(g(x)). This involves substituting g(x) into f(x). Additionally, we need to determine the domain of the resulting function.

Step 2: Substitute g(x) into f(x). Since f(x) = √(x + 4) and g(x) = √(x − 1), replace x in f(x) with g(x). This gives fg(x) = √(g(x) + 4). Substituting g(x), we get fg(x) = √(√(x − 1) + 4).

Step 3: Simplify the expression if possible. In this case, the expression fg(x) = √(√(x − 1) + 4) cannot be simplified further.

Step 4: Determine the domain of fg(x). The domain of a square root function is restricted to values where the expression inside the square root is non-negative. Start by ensuring the inner square root, √(x − 1), is defined. This requires x − 1 ≥ 0, so x ≥ 1. Next, ensure the entire expression √(√(x − 1) + 4) is defined. Since √(x − 1) + 4 is always non-negative for x ≥ 1, no additional restrictions are needed.

Step 5: Combine the domain restrictions. The domain of fg(x) is x ≥ 1. In interval notation, this is [1, ∞).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 evaluating g(x) first and then substituting that result into f(x). Understanding how to properly compose functions is essential for solving the problem.

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 involves solving inequalities to find valid x-values for both f(x) and g(x).

Square Root Functions

Square root functions, such as f(x) = √(x + 4) and g(x) = √(x - 1), are defined only for non-negative inputs. This means that the expressions under the square roots must be greater than or equal to zero. Understanding the behavior and restrictions of square root functions is crucial for finding their domains and composing them correctly.

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