Question

Use the functions below.

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)=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 (−∞,∞)
.
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)=2xxx+7−−−−−√−2x+1x+7−−−−−√
and the domain of (f⋅g⋅h)(x)
is x≥−7
.

Answer 1

To find \( (f \cdot g \cdot h)(x) \), we need to multiply the three functions \( f(x) \), \( g(x) \), and \( h(x) \).

1. **Function Definitions**:
- \( f(x) = x - 2 \)
- \( g(x) = 2x \)
- \( h(x) = \sqrt{x + 7} \)

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

3. **Simplifying**:
\[
= (x - 2)(2x) \cdot \sqrt{x + 7}
\]
\[
= (2x^2 - 4x) \cdot \sqrt{x + 7}
\]

4. **Domain**:
The domain of \( (f \cdot g \cdot h)(x) \) is determined by the components:
- \( f(x) \) and \( g(x) \) are defined for all \( x \in (-\infty, \infty) \).
- \( h(x) = \sqrt{x + 7} \) is defined when \( x + 7 \geq 0 \), or \( x \geq -7 \).

Therefore, the overall domain of \( (f \cdot g \cdot h)(x) \) is \( x \geq -7 \).

### Final Results
\[
(f \cdot g \cdot h)(x) = (2x^2 - 4x) \cdot \sqrt{x + 7}
\]
Domain: \( x \geq -7 \)

So the correct response matches:
- \( (f \cdot g \cdot h)(x) = (2x^2 - 4x) \cdot \sqrt{x + 7} \)
- The domain of \( (f \cdot g \cdot h)(x) \) is \( x \geq -7 \).