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 g(x) is not zero. Understanding these operations is essential for manipulating and analyzing functions.
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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.
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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.
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