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 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, such as f(x) = √(x - 2) and g(x) = √(2 - x), the expressions under the square roots must be non-negative, leading to specific restrictions on x that define the domain.

Square Root Functions

Square root functions, like f(x) = √(x - 2) and g(x) = √(2 - x), are defined only for non-negative values. This means that the expressions inside the square roots must be greater than or equal to zero. Understanding how to manipulate these inequalities is crucial for determining the domain of the functions and ensuring valid outputs.