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)=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 .

(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)=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 .

Answer 1

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

1. \(f(x) = x - 2\)
2. \(g(x) = 2x\)
3. \(h(x) = \sqrt{x + 7}\)

The composition of these functions is given by:

\[ (f \cdot g \cdot h)(x) = f(x) \cdot g(x) \cdot h(x) \]

Calculating step by step:

1. Calculate \(f(x) \cdot g(x)\):

   \[ f(x) \cdot g(x) = (x - 2)(2x) = 2x^2 - 4x \]

2. Now multiply this result by \(h(x)\):

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

So:

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

Next, we determine the domain of \((f \cdot g \cdot h)(x)\). The domain is constrained by the square root function \(h(x) = \sqrt{x + 7}\), which requires:

\[ x + 7 \geq 0 \implies x \geq -7 \]

Thus, the domain of \((f \cdot g \cdot h)(x)\) is:

\[ [x \geq -7 \text{ or } [-7, \infty) \]

In conclusion, we have:

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

\[ x \geq -7 \] (or written in interval notation, \([-7, \infty)\)).