Question

Question

Use the functions below.

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)=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 .

(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)=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)=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 .

Answer 1

To find \((f \cdot g)(x)\), we first express the product of the two functions \(f(x)\) and \(g(x)\):

1. The function \(f(x)\) is given by: \[ f(x) = \sqrt{x} + 2 \]

2. The function \(g(x)\) is given by: \[ g(x) = 8x^3 \]

Now we find the product: \[ (f \cdot g)(x) = f(x) \cdot g(x) = (\sqrt{x} + 2)(8x^3) \]

Distributing \(8x^3\) to both terms in \(f(x)\): \[ (f \cdot g)(x) = 8x^3 \sqrt{x} + 16x^3 \]

Now, simplifying \(8x^3 \sqrt{x}\): \[ 8x^3 \sqrt{x} = 8x^{3 + \frac{1}{2}} = 8x^{\frac{7}{2}} \]

Thus: \[ (f \cdot g)(x) = 8x^{\frac{7}{2}} + 16x^3 \]

Next, we need to determine the domain of \((f \cdot g)(x)\). The domain is affected by the individual domains of \(f(x)\) and \(g(x)\):

• The domain of \(f(x) = \sqrt{x} + 2\) is \(x \geq 0\) (since the square root function is only defined for non-negative values).
• The function \(g(x) = 8x^3\) is defined for all real numbers, or \((-∞, ∞)\).

Since \(f(x)\) limits the domain, the overall domain of \((f \cdot g)(x)\) is: \[ x \geq 0 \quad \text{(or written in interval notation: } [0, \infty)) \]

Thus, the final answers are: \[ (f \cdot g)(x) = 8x^{\frac{7}{2}} + 16x^3 \] And the domain is: \[ x \geq 0 \]

From the options provided, the closest correct response is: \[ (f \cdot g)(x) = 8x + 48x^2\sqrt{3} + 96\sqrt{x} + 64 \quad \text{ and the domain of } (f \cdot g)(x) \text{ is } x \geq 0. \]

However, the actual expression for \((f \cdot g)(x)\) should be corrected to match the work shown above.