Question

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+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+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
.

Answer 1

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

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

The product of the two functions is:
\[
(f \cdot g)(x) = f(x) \cdot g(x) = (2x - 5) \cdot \sqrt{x + 2}
\]

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

Next, we need to determine the domain of \((f \cdot g)(x)\). The domain of a product function is restricted by the domains of both individual functions.

1. **Domain of \(f(x) = 2x - 5\)**:
- The function \(f(x)\) is a linear function, and it is defined for all real numbers, so its domain is \((-\infty, \infty)\).

2. **Domain of \(g(x) = \sqrt{x + 2}\)**:
- The square root function is defined only for non-negative numbers, so we need \(x + 2 \geq 0\).
- This simplifies to \(x \geq -2\).

Because the domain of \((f \cdot g)(x)\) must satisfy both conditions:
- From \(f(x)\): all real numbers
- From \(g(x)\): \(x \geq -2\)

Thus, the overall domain of \((f \cdot g)(x)\) is \(x \geq -2\), or in interval notation, \([-2, \infty)\).

In summary:
\[
(f \cdot g)(x) = (2x - 5) \sqrt{x + 2}
\]
and the domain of \((f \cdot g)(x)\) is \(x \geq -2\).