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: Find ƒ+g. The sum of the two functions is given by (ƒ+g)(x) = f(x) + g(x). Substituting the given functions, we have (ƒ+g)(x) = √(x - 2) + √(2 - x). To determine the domain, both square roots must be defined, meaning x - 2 ≥ 0 and 2 - x ≥ 0. Solve these inequalities to find the domain.
3. Step 3: Find ƒ−g. The difference of the two functions is given by (ƒ−g)(x) = f(x) - g(x). Substituting the given functions, we have (ƒ−g)(x) = √(x - 2) - √(2 - x). The domain is the same as for ƒ+g, as it depends on the square roots being defined. Use the same inequalities to determine the domain.
4. Step 4: Find ƒg. The product of the two functions is given by (ƒg)(x) = f(x) * g(x). Substituting the given functions, we have (ƒg)(x) = √(x - 2) * √(2 - x). The domain is again determined by ensuring both square roots are defined, so solve the inequalities x - 2 ≥ 0 and 2 - x ≥ 0.
5. Step 5: Find ƒ/g. The quotient of the two functions is given by (ƒ/g)(x) = f(x) / g(x). Substituting the given functions, we have (ƒ/g)(x) = √(x - 2) / √(2 - x). In addition to ensuring the square roots are defined, we must also ensure the denominator is not zero, meaning √(2 - x) ≠ 0. Solve the inequalities and consider this additional restriction to determine the domain.

Key Concepts

Function Operations

Function operations involve combining two functions through addition, subtraction, multiplication, or division. For functions f and g, these 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 that g(x) is not zero.

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 solving inequalities to find the valid x-values for each function.

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 inside the square roots must be greater than or equal to zero, which directly influences the domain of the functions and the results of the operations performed on them.