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Use the functions below.

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−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
.
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 x is greater than or equal to negative 2 .

(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+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 .

1 answer

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

Given:

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

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

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

So the expression for \((f \cdot g)(x)\) is: \[ (f \cdot g)(x) = (2x - 5) \sqrt{x + 2} \]

Next, we need to find the domain of \((f \cdot g)(x)\).

  1. The function \(\sqrt{x + 2}\) is defined when the expression under the square root is non-negative: \[ x + 2 \geq 0 \implies x \geq -2 \]

  2. The function \(2x - 5\) is a linear function and is defined for all real numbers.

Therefore, the domain of \((f \cdot g)(x)\) is determined by the condition from \(g(x)\) (the square root), which is: \[ \text{Domain: } x \geq -2 \]

In interval notation, this is written as \([-2, \infty)\).

Putting it all together, the answers are:

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

Since none of the provided responses match exactly, you might want to choose the closest option based on the domain requirement, which is \(x \geq -2\).