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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Step 1: To find \( f+g \), add the functions: \( f(x) + g(x) = \sqrt{x-2} + \sqrt{2-x} \). Determine the domain by finding the intersection of the domains of \( f(x) \) and \( g(x) \).

Step 2: To find \( f-g \), subtract the functions: \( f(x) - g(x) = \sqrt{x-2} - \sqrt{2-x} \). The domain is the same as for \( f+g \), which is the intersection of the domains of \( f(x) \) and \( g(x) \).

Step 3: To find \( fg \), multiply the functions: \( f(x) \cdot g(x) = \sqrt{x-2} \cdot \sqrt{2-x} \). The domain is the intersection of the domains of \( f(x) \) and \( g(x) \).

Step 4: To find \( \frac{f}{g} \), divide the functions: \( \frac{f(x)}{g(x)} = \frac{\sqrt{x-2}}{\sqrt{2-x}} \). The domain is the intersection of the domains of \( f(x) \) and \( g(x) \), excluding any values that make \( g(x) = 0 \).

Step 5: Determine the domain of each function. For \( f(x) = \sqrt{x-2} \), the domain is \( x \geq 2 \). For \( g(x) = \sqrt{2-x} \), the domain is \( x \leq 2 \). The intersection of these domains is \( x = 2 \). For \( \frac{f}{g} \), ensure \( g(x) \neq 0 \), so the domain is empty.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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