Question

Consider the following. 1 f(x) = 9(x) = x + 2 (a) Find the function (f. g)(x). 1 √x+2 (f. g)(x) = Find the domain of (f. g)(x). (Enter your answer using interval notation.) (b) Find the function (g. f)(x). (g.f)(x) = (1+2√x) VX Find the domain of (g. 1)(x). (Enter your answer using interval notation.) (c) Find the function (f.)(x). (8.5(x) Find the domain of (f. f)(x). (Enter your answer using interval notation.)

Answer 1

(a) The function
`\((f \cdot g)(x) = (1)/(√(x + 2))\)`. The domain is
`\(x \in (-2, \infty)\)`. (b) The function
`\((g \cdot f)(x) = (1 + 2√(x))√(x)\)`. The domain is
`\(x \in [0, \infty)\)`. (c) The function
`\((f \cdot f)(x) = 8.5(x)\)`. The domain is
`\(x \in (-\infty, \infty)\)`.

Step-by-step explanation:

(a) To find
`\((f \cdot g)(x)\)`, we multiply the two functions
`\(f(x) = 9\) and \(g(x) = (1)/(√(x + 2))\)`. The result is
`\((f \cdot g)(x) = (1)/(√(x + 2))\).` To determine the domain, we consider that the expression under the square root must be greater than zero, leading to x + 2 > 0 and x > -2.

(b) For
`\((g \cdot f)(x)\)` the functions
`\(g(x) = (1)/(√(x + 2))\)` and f(x) = 9are multiplied. The outcome is
`\((g \cdot f)(x) = (1 + 2√(x))√(x)\)`. The domain is determined by the requirement that the expression under the square root must be greater than or equal to zero, resulting in
`\(x \geq 0\)`.

(c) The function
`\((f \cdot f)(x)\)` is obtained by squaring the function
`\(f(x) = 8.5\)\\`The result is
`\((f \cdot f)(x) = 8.5x\)`, and the domain is the set of all real numbers.