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

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the function (f - g)(x), which is the difference between the functions f(x) and g(x).
2. Step 2: Write the expression for (f - g)(x). This is given by (f - g)(x) = f(x) - g(x).
3. Step 3: Substitute the given functions into the expression. We have f(x) = \sqrt{x} and g(x) = x - 4, so (f - g)(x) = \sqrt{x} - (x - 4).
4. Step 4: Simplify the expression. Distribute the negative sign: (f - g)(x) = \sqrt{x} - x + 4.
5. Step 5: Determine the domain of (f - g)(x). The domain of f(x) = \sqrt{x} is x \geq 0, and the domain of g(x) = x - 4 is all real numbers. The domain of (f - g)(x) is the intersection of these domains, which is x \geq 0.

Key Concepts

Function Operations

Function operations involve combining two functions to create a new function. In this case, f(x) and g(x) can be combined through subtraction to find f - g. Understanding how to perform operations on functions is essential for manipulating and analyzing their behavior.

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 f(x) = √x, the domain is x ≥ 0, since the square root of a negative number is not defined in the real number system. Identifying the domain is crucial when performing operations on functions to ensure the resulting function is valid.

Composite Functions

Composite functions are formed when one function is applied to the result of another function. In this context, after finding f - g, it is important to analyze how the domains of f and g interact. Understanding composite functions helps in determining the overall domain of the resulting function after operations are performed.