Question

The length of a rectangular garden is given by the function f(x)=x+4
. The width of a rectangular garden is given by the function g(x)=2x−1
. What function represents the area of the rectangular garden? What is the domain?(1 point)
Responses

(f⋅g)(x)=3x+3
and the domain of (f⋅g)(x)
is (−∞,∞)
.
open paren f times g close paren of x is equal to 3 x plus 3 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)=3x+3
and the domain of (f⋅g)(x)
is x≥12
.
open paren f times g close paren of x is equal to 3 x plus 3 and the domain of open paren f times g close paren of x is x is greater than or equal to 1 half .

(f⋅g)(x)=2x2+7x−4
and the domain of (f⋅g)(x)
is (−∞,∞)
.
open paren f times g close paren of x is equal to 2 x squared plus 7 x minus 4 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)=2x2+7x−4
and the domain of (f⋅g)(x)
is x≥12
.
open paren f times g close paren of x is equal to 2 x squared plus 7 x minus 4 and the domain of open paren f times g close paren of x is x is greater than or equal to 1 half .
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Answer 1

To find the function that represents the area of the rectangular garden, we need to multiply the length function \( f(x) \) by the width function \( g(x) \).

Given:
- Length \( f(x) = x + 4 \)
- Width \( g(x) = 2x - 1 \)

The area \( A(x) \) is given by the product:
\[
A(x) = f(x) \cdot g(x) = (x + 4)(2x - 1)
\]

Now, we'll expand this product:
\[
A(x) = (x + 4)(2x - 1) = x \cdot 2x + x \cdot (-1) + 4 \cdot 2x + 4 \cdot (-1)
\]
\[
= 2x^2 - x + 8x - 4
\]
\[
= 2x^2 + 7x - 4
\]

Therefore, the function that represents the area of the rectangular garden is:
\[
(f \cdot g)(x) = 2x^2 + 7x - 4
\]

Next, we determine the domain of this function. Since both the length and width functions \( f(x) \) and \( g(x) \) must be greater than zero for the area to make sense (i.e., the garden cannot have negative or zero dimensions), we set up the inequalities:

1. For \( f(x) > 0 \):
\[
x + 4 > 0 \implies x > -4
\]

2. For \( g(x) > 0 \):
\[
2x - 1 > 0 \implies 2x > 1 \implies x > \frac{1}{2}
\]

The more restrictive condition is \( x > \frac{1}{2} \). Thus, the domain of \( (f \cdot g)(x) \) is:
\[
x \geq \frac{1}{2}
\]

**Final response:**
\[
(f \cdot g)(x) = 2x^2 + 7x - 4 \text{ and the domain is } x \geq \frac{1}{2}.
\]