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 f(x) = √(x - 2) and g(x) = √(2 - x). Additionally, we need to determine the domain for each resulting function. Step 2: Start with the sum (ƒ+g). The sum of two functions is defined as (ƒ+g)(x) = f(x) + g(x). Substitute the given functions: (ƒ+g)(x) = √(x - 2) + √(2 - x). To determine the domain, ensure that both square roots are defined, which means the expressions inside the square roots must be non-negative. Solve x - 2 ≥ 0 and 2 - x ≥ 0 to find the intersection of their domains. Step 3: Move to the difference (ƒ−g). The difference of two functions is defined as (ƒ−g)(x) = f(x) - g(x). Substitute the given functions: (ƒ−g)(x) = √(x - 2) - √(2 - x). The domain for this function is the same as the domain for the sum, as it depends on the same square root expressions. Use the results from Step 2 to determine the domain. Step 4: Find the product (ƒg). The product of two functions is defined as (ƒg)(x) = f(x) * g(x). Substitute the given functions: (ƒg)(x) = √(x - 2) * √(2 - x). The domain is again determined by ensuring that both square roots are defined, which means solving x - 2 ≥ 0 and 2 - x ≥ 0, as in Step 2. Step 5: Determine the quotient (ƒ/g). The quotient of two functions is defined as (ƒ/g)(x) = f(x) / g(x), provided g(x) ≠ 0. Substitute the given functions: (ƒ/g)(x) = √(x - 2) / √(2 - x). The domain is determined by ensuring that both square roots are defined (x - 2 ≥ 0 and 2 - x \u003e 0, note the strict inequality for the denominator). Solve these inequalities to find the domain.

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)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Function Operations

Domain of a Function

Square Root Functions