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

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the composition of two functions, denoted as fg, which means f(g(x)).
2. Step 2: Substitute g(x) into f(x). Since f(x) = \sqrt{x + 4} and g(x) = \sqrt{x - 1}, we substitute g(x) into f(x) to get f(g(x)) = \sqrt{\sqrt{x - 1} + 4}.
3. Step 3: Simplify the expression if possible. In this case, the expression \sqrt{\sqrt{x - 1} + 4} is already simplified.
4. Step 4: Determine the domain of the composition function fg. The domain of fg is determined by the domains of both f(x) and g(x).
5. Step 5: Find the domain restrictions. For g(x) = \sqrt{x - 1}, x must be greater than or equal to 1. For f(x) = \sqrt{x + 4}, the input must be greater than or equal to -4. However, since we are substituting g(x) into f(x), we need \sqrt{x - 1} + 4 \geq 0, which simplifies to x \geq 1. Therefore, the domain of fg is x \geq 1.

Key Concepts

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 the 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.