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 functions f(x) = \sqrt{x - 2} and g(x) = \sqrt{2 - x}.
2. Step 2: To find (f+g)(x), add the functions: (f+g)(x) = \sqrt{x - 2} + \sqrt{2 - x}.
3. Step 3: To find (f-g)(x), subtract the functions: (f-g)(x) = \sqrt{x - 2} - \sqrt{2 - x}.
4. Step 4: To find (fg)(x), multiply the functions: (fg)(x) = \sqrt{x - 2} \cdot \sqrt{2 - x}.
5. Step 5: To find (f/g)(x), divide the functions: (f/g)(x) = \frac{\sqrt{x - 2}}{\sqrt{2 - x}}. Determine the domain for each function by finding the values of x that make the expressions under the square roots non-negative.

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. Understanding the behavior of these functions is crucial for determining their domains and performing operations on them.