Question

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)

Accepted Answer

Step 1: Understand the problem. We are tasked with finding the sum (ƒ+g), difference (ƒ−g), product (ƒg), and quotient (ƒ/g) of the two functions ƒ(x) = √(x - 2) and g(x) = √(2 - x). Additionally, we need to determine the domain for each resulting function. Step 2: Find ƒ+g. The sum of the functions is given by (ƒ+g)(x) = ƒ(x) + g(x). Substitute the given functions: (ƒ+g)(x) = √(x - 2) + √(2 - x). To determine the domain, ensure that the expressions inside both square roots are non-negative. Solve x - 2 ≥ 0 and 2 - x ≥ 0 to find the intersection of their valid intervals. Step 3: Find ƒ−g. The difference of the functions is given by (ƒ−g)(x) = ƒ(x) - g(x). Substitute the given functions: (ƒ−g)(x) = √(x - 2) - √(2 - x). The domain is the same as for ƒ+g, as it depends on the same square root expressions being defined. Step 4: Find ƒg. The product of the functions is given by (ƒg)(x) = ƒ(x) * g(x). Substitute the given functions: (ƒg)(x) = √(x - 2) * √(2 - x). Simplify the product using the property of square roots: √(x - 2) * √(2 - x) = √((x - 2)(2 - x)). The domain is determined by ensuring the argument of the square root, (x - 2)(2 - x), is non-negative. Step 5: Find ƒ/g. The quotient of the functions is given by (ƒ/g)(x) = ƒ(x) / g(x). Substitute the given functions: (ƒ/g)(x) = √(x - 2) / √(2 - x). The domain is determined by ensuring both square roots are defined (as in previous steps) and that the denominator, √(2 - x), is not zero. Solve 2 - x ≠ 0 to exclude any values that make the denominator zero.

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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Concetti chiave

Function Operations

Domain of a Function

Square Root Functions