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)

Verified step by step guidance

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 ƒ(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 functions is given by (ƒ+g)(x) = ƒ(x) + g(x). Substitute the given functions: (ƒ+g)(x) = √(x - 2) + √(2 - x). To determine the domain, ensure that the expressions inside both square roots are non-negative. Solve x - 2 ≥ 0 and 2 - x ≥ 0 to find the intersection of their valid intervals.
3. Step 3: Find ƒ−g. The difference of the functions is given by (ƒ−g)(x) = ƒ(x) - g(x). Substitute the given functions: (ƒ−g)(x) = √(x - 2) - √(2 - x). The domain is the same as for ƒ+g, as it depends on the same square root expressions being defined.
4. Step 4: Find ƒg. The product of the functions is given by (ƒg)(x) = ƒ(x) * g(x). Substitute the given functions: (ƒg)(x) = √(x - 2) * √(2 - x). Simplify the product using the property of square roots: √(x - 2) * √(2 - x) = √((x - 2)(2 - x)). The domain is determined by ensuring the argument of the square root, (x - 2)(2 - x), is non-negative.
5. Step 5: Find ƒ/g. The quotient of the functions is given by (ƒ/g)(x) = ƒ(x) / g(x). Substitute the given functions: (ƒ/g)(x) = √(x - 2) / √(2 - x). The domain is determined by ensuring both square roots are defined (as in previous steps) and that the denominator, √(2 - x), is not zero. Solve 2 - x ≠ 0 to exclude any values that make the denominator zero.

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 inputs. This means that the expressions under the square roots must be greater than or equal to zero, which directly influences the domain of the functions and any operations performed on them.