Find ƒ+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. You are tasked with finding the sum of two functions, ƒ(x) and g(x), which is denoted as (ƒ + g)(x). This means you need to add the two functions together: (ƒ + g)(x) = ƒ(x) + g(x).
2. Step 2: Write the expressions for ƒ(x) and g(x). Here, ƒ(x) = √x and g(x) = x − 4. Substitute these into the formula for (ƒ + g)(x): (ƒ + g)(x) = √x + (x − 4).
3. Step 3: Simplify the expression if possible. In this case, the expression (ƒ + g)(x) = √x + x − 4 cannot be simplified further because the square root and linear terms are not like terms.
4. Step 4: Determine the domain of the resulting function. The domain is the set of all x-values for which the function is defined. For ƒ(x) = √x, the square root requires x ≥ 0 (since you cannot take the square root of a negative number in the real number system). For g(x) = x − 4, there are no restrictions on x. Therefore, the domain of (ƒ + g)(x) is x ≥ 0.
5. Step 5: Write the final domain in interval notation. Since x must be greater than or equal to 0, the domain is [0, ∞).

Key Concepts

Function Addition

Function addition involves combining two functions, f(x) and g(x), to create a new function, denoted as (f + g)(x) = f(x) + g(x). In this case, you will add the outputs of f(x) and g(x) for each input x, resulting in a new expression that represents the sum of the two functions.

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 functions f(x) = √x and g(x) = x - 4, you must consider the restrictions imposed by each function, particularly the square root function, which requires non-negative inputs.

Combining Domains

When adding two functions, the domain of the resulting function (f + g) is determined by the intersection of the individual domains of f and g. This means you need to find the common x-values that satisfy the conditions for both functions, ensuring that the combined function is valid across its entire domain.