Use the functions below.

Question

f(x)=2x−5
g(x)=x+2−−−−√
What is (f⋅g)(x)
? What is the domain of (f⋅g)(x)
?

(1 point)
Responses

(f⋅g)(x)=2x+2√−5
and the domain of (f⋅g)(x)
is (−∞, ∞)
.
open paren f times g close paren of x is equal to 2 raised to the square root of x plus 2 end root power minus 5 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−3−−−−−√
and the domain of (f⋅g)(x)
is (−∞, ∞)
.
open paren f times g close paren of x is equal to square root of 2 to the x th power minus 3 end root 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−3−−−−−√
and the domain of (f⋅g)(x)
is x≥−2
.
open paren f times g close paren of x is equal to square root of 2 to the x th power minus 3 end root and the domain of open paren f times g close paren of x is x is greater than or equal to negative 2 .

(f⋅g)(x)=2x+2√−5
and the domain of (f⋅g)(x)
is x≥−2
.

Answer 1

To find \((f \cdot g)(x)\), we multiply the functions \(f(x)\) and \(g(x)\):

1. Given functions:

   • \(f(x) = 2x - 5\)
   • \(g(x) = \sqrt{x + 2}\)

2. Multiply \(f(x)\) and \(g(x)\): \[ (f \cdot g)(x) = f(x) \cdot g(x) = (2x - 5) \cdot \sqrt{x + 2} \]

So, we have: \[ (f \cdot g)(x) = (2x - 5) \sqrt{x + 2} \]

3. Finding the domain: The domain of \((f \cdot g)(x)\) is restricted by both \(f(x)\) and \(g(x)\). The function \(g(x) = \sqrt{x + 2}\) requires that the expression inside the square root be non-negative, which leads to: \[ x + 2 \geq 0 \quad \Rightarrow \quad x \geq -2 \] The function \(f(x)\) (a linear function) is defined for all real numbers, so it does not impose additional restrictions.

Therefore, the domain of \((f \cdot g)(x)\) is: \[ x \geq -2 \quad \text{or in interval notation, } [-2, \infty) \]

Putting it all together:

\((f \cdot g)(x) = (2x - 5) \sqrt{x + 2}\) and the domain of \((f \cdot g)(x)\) is \(x \geq -2\) or \([-2, \infty)\).

The correct response based on the given options appears to be: \((f \cdot g)(x) = (2x - 5) \sqrt{x + 2}\) and the domain of \((f \cdot g)(x)\) is \(x \geq -2\).