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: Find ƒ+g. The sum of the two functions is given by (ƒ+g)(x) = f(x) + g(x). Substituting the given functions, we have (ƒ+g)(x) = √(x - 2) + √(2 - x). To determine the domain, both square roots must be defined, meaning x - 2 ≥ 0 and 2 - x ≥ 0. Solve these inequalities to find the domain. Step 3: Find ƒ−g. The difference of the two functions is given by (ƒ−g)(x) = f(x) - g(x). Substituting the given functions, we have (ƒ−g)(x) = √(x - 2) - √(2 - x). The domain is the same as for ƒ+g, as it depends on the square roots being defined. Use the same inequalities to determine the domain. Step 4: Find ƒg. The product of the two functions is given by (ƒg)(x) = f(x) * g(x). Substituting the given functions, we have (ƒg)(x) = √(x - 2) * √(2 - x). The domain is again determined by ensuring both square roots are defined, so solve the inequalities x - 2 ≥ 0 and 2 - x ≥ 0. Step 5: Find ƒ/g. The quotient of the two functions is given by (ƒ/g)(x) = f(x) / g(x). Substituting the given functions, we have (ƒ/g)(x) = √(x - 2) / √(2 - x). In addition to ensuring the square roots are defined, we must also ensure the denominator is not zero, meaning √(2 - x) ≠ 0. Solve the inequalities and consider this additional restriction to determine 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)

Risposta video verificata per un problema simile:

Concetti chiave

Function Operations

Domain of a Function

Square Root Functions