Use the functions below.

Question

f(x)=x−−√3+2
g(x)=8x3
What is (f⋅g)(x)
? What is the domain of (f⋅g)(x)
?

(1 point)
Responses

(f⋅g)(x)=8x+48x2−−√3+96x−−√3+64
and the domain of (f⋅g)(x)
is x≥0
.
open paren f times g close paren of x is equal to 8 x plus 48 cube root of x squared plus 96 cube root of x plus 64 and the domain of open paren f times g close paren of x is x is greater than or equal to 0 .

(f⋅g)(x)=8x+48x2−−√3+96x−−√3+64
and the domain of (f⋅g)(x)
is (−∞, ∞)
.
open paren f times g close paren of x is equal to 8 x plus 48 cube root of x squared plus 96 cube root of x plus 64 and the domain of open paren f times g close paren of x is open paren negative infinity comma infinity close paren .

(f⋅g)(x)=2x+2
and the domain of (f⋅g)(x)
is x≥0
.
open paren f times g close paren of x is equal to 2 x plus 2 and the domain of open paren f times g close paren of x is x is greater than or equal to 0 .

(f⋅g)(x)=2x+2
and the domain of (f⋅g)(x)
is (−∞, ∞)
.
open paren f times g close paren of x is equal to 2 x plus 2 and the domain of open paren f times g close paren of x is open paren negative infinity comma infinity close paren .

Answer 1

To find \((f \cdot g)(x)\), we start by calculating the product of the functions \(f(x)\) and \(g(x)\).

1. Define the Functions:

   • \( f(x) = \sqrt{x} + 2 \)
   • \( g(x) = 8x^3 \)

2. Calculate \((f \cdot g)(x)\): \[ (f \cdot g)(x) = f(x) \cdot g(x) = (\sqrt{x} + 2)(8x^3) \]

3. Distribute: \[ (f \cdot g)(x) = 8x^3 \cdot \sqrt{x} + 8x^3 \cdot 2 \] \[ = 8x^{3.5} + 16x^3 \] \[ = 8x^{7/2} + 16x^3 \]

4. Domain Determination:

   • The domain of \( f(x) = \sqrt{x} + 2 \) is \( x \geq 0 \) because the square root function is defined only for non-negative values of \( x \).
   • The expression \( g(x) = 8x^3 \) is defined for all real numbers \( x \).
   • Therefore, the domain of \((f \cdot g)(x)\) is determined by the more restrictive domain of \( f(x) \), which is \( x \geq 0 \).

5. Final Result:

   • \((f \cdot g)(x) = 8x^{7/2} + 16x^3\)
   • The domain of \((f \cdot g)(x)\) is \( x \geq 0 \).

Thus, the correct statement is: \[ (f \cdot g)(x) = 8x^{7/2} + 16x^3 \quad \text{and the domain of } (f \cdot g)(x) \text{ is } x \geq 0. \]