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)

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 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, such as f(x) = √(x - 2) and g(x) = √(2 - x), the expressions under the square roots must be non-negative. Therefore, determining the domain requires solving inequalities to find the valid x-values that keep the function outputs real.

Square Root Functions

Square root functions, like f(x) = √(x - 2) and g(x) = √(2 - x), are defined only for non-negative inputs. The function f(x) is defined when x - 2 ≥ 0, leading to x ≥ 2, while g(x) is defined when 2 - x ≥ 0, leading to x ≤ 2. Understanding the behavior and restrictions of square root functions is crucial for correctly determining their domains and performing operations on them.