In Exercises 82–84, find f + g, f - g, fg, and f/g. Determine the domain for each function. f(x) = √(x + 7), g(x) = √(x - 2)

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the sum (f + g), difference (f - g), product (fg), and quotient (f/g) of the functions f(x) = √(x + 7) and g(x) = √(x - 2). Additionally, you need to determine the domain for each resulting function.
2. Step 2: To find f + g, add the two functions: (f + g)(x) = √(x + 7) + √(x - 2). The domain of this function is determined by the intersection of the domains of f(x) and g(x). For f(x), the domain is x ≥ -7 (since the square root requires non-negative values). For g(x), the domain is x ≥ 2. The intersection of these domains is x ≥ 2.
3. Step 3: To find f - g, subtract the two functions: (f - g)(x) = √(x + 7) - √(x - 2). The domain is the same as for f + g, which is x ≥ 2, because the square root functions must remain defined.
4. Step 4: To find fg, multiply the two functions: (fg)(x) = √(x + 7) * √(x - 2). The domain is again the intersection of the domains of f(x) and g(x), which is x ≥ 2.
5. Step 5: To find f/g, divide the two functions: (f/g)(x) = √(x + 7) / √(x - 2). In addition to the domain restrictions from f(x) and g(x), you must exclude any x-values that make the denominator zero. Since g(x) = √(x - 2), the denominator is zero when x = 2. Therefore, the domain of f/g is x > 2.

Key Concepts

Function Operations

Function operations involve combining two functions through addition, subtraction, multiplication, or division. For example, if f(x) and g(x) are two functions, f + g means adding their outputs, while f - g means subtracting the output of g from f. Understanding these operations is essential for manipulating and analyzing functions in algebra.

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, like f(x) = √(x + 7) and g(x) = √(x - 2), the expressions under the square roots must be non-negative. Identifying the domain is crucial for ensuring that the operations performed on the functions yield valid results.

Composite Functions

Composite functions are formed when one function is applied to the result of another function. In the context of the given functions f and g, operations like f/g require understanding how the output of g affects the input of f. This concept is important for determining the overall behavior and restrictions of the resulting function, especially regarding its domain.