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

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Step 1: Understand the problem. We are tasked with finding the difference of two functions, denoted as (f − g)(x), which is defined as f(x) − g(x). Additionally, we need to determine the domain of the resulting function.

Step 2: Write the expression for (f − g)(x). Substitute the given functions f(x) = √x and g(x) = x − 4 into the formula for (f − g)(x): (f − g)(x) = f(x) − g(x) = √x − (x − 4).

Step 3: Simplify the expression for (f − g)(x). Distribute the negative sign across the terms in g(x): (f − g)(x) = √x − x + 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 f(x) = √x, the square root requires x ≥ 0. For g(x) = x − 4, there are no restrictions. Therefore, the domain of (f − g)(x) is x ≥ 0.

Step 5: Express the domain in interval notation. Since x must be greater than or equal to 0, the domain is [0, ∞).

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Key Concepts

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

Function Operations

Function operations involve combining two functions to create a new function. In this case, f−g means subtracting the function g(x) from f(x). 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, as square roots of negative numbers are not defined in the real number system. When combining functions, the domain of the resulting function must consider the domains of both original functions.

Piecewise Functions

Piecewise functions are defined by different expressions based on the input value. When finding f−g, the resulting function may need to be expressed piecewise if the domains of f and g do not overlap. This concept is crucial for accurately determining the domain of the combined function.

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