The length of a rectangular garden is given by the function f(x)=x+4

Question

. 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)=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 .

(f⋅g)(x)=3x+3
and the domain of (f⋅g)(x)
is x≥12
.

Answer 1

To find the function that represents the area of the rectangular garden given the length \( f(x) = x + 4 \) and the width \( g(x) = 2x - 1 \), we calculate the area \( A(x) \) by multiplying the two functions:

\[ A(x) = f(x) \cdot g(x) = (x + 4)(2x - 1) \]

Now we can expand this expression:

\[ A(x) = (x + 4)(2x - 1) = x(2x) + x(-1) + 4(2x) + 4(-1) = 2x^2 - x + 8x - 4 \] \[ A(x) = 2x^2 + 7x - 4 \]

Next, we need to determine the domain for the area function \( A(x) \). Since the length and width must be non-negative:

1. For the length \( f(x) = x + 4 \): \[ x + 4 \geq 0 \implies x \geq -4 \]

2. For the width \( g(x) = 2x - 1 \): \[ 2x - 1 \geq 0 \implies 2x \geq 1 \implies x \geq \frac{1}{2} \]

The more restrictive condition is \( x \geq \frac{1}{2} \).

Thus, the area function is: \[ A(x) = 2x^2 + 7x - 4 \]

And the domain of \( A(x) \) is: \[ x \geq \frac{1}{2} \]

So the correct response is: (f⋅g)(x)=2x²+7x−4 and the domain of (f⋅g)(x) is \( x \geq \frac{1}{2} \).