In Exercises 31–50, find ƒ+g, f−g, fg, and f/g. Determine the domain for each function. f(x) = √(x -2), g(x) = √(2-x)

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1. Step 1: Understand the problem. We are tasked with finding the sum (ƒ+g), difference (ƒ−g), product (ƒg), and quotient (ƒ/g) of the two functions f(x) = √(x - 2) and g(x) = √(2 - x). Additionally, we need to determine the domain for each resulting function.
2. Step 2: Start with the sum (ƒ+g). The sum of two functions is defined as (ƒ+g)(x) = f(x) + g(x). Substitute the given functions: (ƒ+g)(x) = √(x - 2) + √(2 - x). To determine the domain, ensure that both square roots are defined, which means the expressions inside the square roots must be non-negative. Solve x - 2 ≥ 0 and 2 - x ≥ 0 to find the intersection of their domains.
3. Step 3: Move to the difference (ƒ−g). The difference of two functions is defined as (ƒ−g)(x) = f(x) - g(x). Substitute the given functions: (ƒ−g)(x) = √(x - 2) - √(2 - x). The domain for this function is the same as the domain for the sum, as it depends on the same square root expressions. Use the results from Step 2 to determine the domain.
4. Step 4: Find the product (ƒg). The product of two functions is defined as (ƒg)(x) = f(x) * g(x). Substitute the given functions: (ƒg)(x) = √(x - 2) * √(2 - x). The domain is again determined by ensuring that both square roots are defined, which means solving x - 2 ≥ 0 and 2 - x ≥ 0, as in Step 2.
5. Step 5: Determine the quotient (ƒ/g). The quotient of two functions is defined as (ƒ/g)(x) = f(x) / g(x), provided g(x) ≠ 0. Substitute the given functions: (ƒ/g)(x) = √(x - 2) / √(2 - x). The domain is determined by ensuring that both square roots are defined (x - 2 ≥ 0 and 2 - x > 0, note the strict inequality for the denominator). Solve these inequalities to find the domain.

Key Concepts

Function Operations

Function operations involve combining two functions through addition, subtraction, multiplication, or division. For functions f and g, the operations are defined as (f + g)(x) = f(x) + g(x), (f - g)(x) = f(x) - g(x), (fg)(x) = f(x) * g(x), and (f/g)(x) = f(x) / g(x), provided g(x) is not zero. Understanding these operations is essential for manipulating and analyzing 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 functions involving square roots, the expression inside the root must be non-negative. Therefore, determining the domain requires identifying the values of x that satisfy these conditions, ensuring that the function outputs real numbers.

Square Root Functions

Square root functions, such as f(x) = √(x - 2) and g(x) = √(2 - x), are defined only for non-negative arguments. This means that the expressions under the square root must be greater than or equal to zero. Understanding the behavior of these functions is crucial for determining their domains and for performing operations like addition and subtraction, which may introduce additional restrictions.