Use the functions below.

Question

f(x)=x−2
g(x)=2x
h(x)=x+7−−−−√
What is (f⋅g⋅h)(x)
? What is the domain of (f⋅g⋅h)(x)
?

(1 point)
Responses

(f⋅g⋅h)(x)=x−2+2x+x+7−−−−−√
and the domain of (f⋅g⋅h)(x)
is (−∞,∞)
.
open paren f times g times h close paren times x is equal to x minus 2 plus 2 to the x th power plus square root of x plus 7 end root and the domain of open paren f times g times h close paren times x is open paren negative infinity comma infinity close paren .

(f⋅g⋅h)(x)=2xxx+7−−−−−√−2x+1x+7−−−−−√
and the domain of (f⋅g⋅h)(x)
is (−∞,∞)
.
open paren f times g times h close paren times x is equal to 2 to the x th power x square root of x plus 7 end root minus 2 raised to the x plus 1 power square root of x plus 7 end root and the domain of open paren f times g times h close paren times x is open paren negative infinity comma infinity close paren .

(f⋅g⋅h)(x)=x−2+2x+x+7−−−−−√
and the domain of (f⋅g⋅h)(x)
is x≥−7
.
open paren f times g times h close paren times x is equal to x minus 2 plus 2 to the x th power plus square root of x plus 7 end root and the domain of open paren f times g times h close paren times x is x is greater than or equal to negative 7 .

(f⋅g⋅h)(x)=2xxx+7−−−−−√−2x+1x+7−−−−−√
and the domain of (f⋅g⋅h)(x)
is x≥−7
.
open paren f times g times h close paren times x is equal to 2 to the x th power x square root of x plus 7 end root minus 2 raised to the x plus 1 power square root of x plus 7 end root and the domain of open paren f times g times h close paren times x is x is greater than or equal to negative 7 .

Answer 1

To find \((f \cdot g \cdot h)(x)\), we first need to determine how these functions will combine through multiplication.

Given:

• \(f(x) = x - 2\)
• \(g(x) = 2x\)
• \(h(x) = \sqrt{x + 7}\)

The product \((f \cdot g \cdot h)(x)\) is calculated as follows:

\[ (f \cdot g \cdot h)(x) = f(x) \cdot g(x) \cdot h(x) = (x - 2) \cdot (2x) \cdot \sqrt{x + 7} \]

Now, breaking this down:

1. First, multiply \(f(x)\) and \(g(x)\): \[ f(x) \cdot g(x) = (x - 2)(2x) = 2x^2 - 4x \]

2. Then, multiply that result by \(h(x)\): \[ (2x^2 - 4x) \cdot \sqrt{x + 7} \]

So, we have: \[ (f \cdot g \cdot h)(x) = (2x^2 - 4x) \cdot \sqrt{x + 7} \]

This would not simplify further into a polynomial, since \(\sqrt{x + 7}\) is not a polynomial term.

Next, we need to determine the domain of \((f \cdot g \cdot h)(x)\):

1. The function \(h(x) = \sqrt{x + 7}\) requires that: \[ x + 7 \geq 0 \implies x \geq -7 \]

2. The functions \(f\) and \(g\) are polynomials and are defined for all real numbers.

Thus, the overall domain of \((f \cdot g \cdot h)(x)\) is limited by the square root function:

\[ \text{Domain} = [-7, \infty) \]

Consequently, combining these results:

\[ (f \cdot g \cdot h)(x) = (2x^2 - 4x) \sqrt{x + 7} \quad \text{and the domain is } x \geq -7. \]