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)=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)=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

Answer 1

To find the area of the rectangular garden, we can multiply the functions that represent the length and width of the garden.

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

The area \( A(x) \) of the rectangle can be expressed as:
\[
A(x) = f(x) \cdot g(x) = (x + 4)(2x - 1)
\]

Now, let's compute this product:
\[
A(x) = (x + 4)(2x - 1) = x(2x - 1) + 4(2x - 1)
\]
\[
= 2x^2 - x + 8x - 4
\]
\[
= 2x^2 + 7x - 4
\]

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

Next, let's determine the domain of this function. Since the length and width represent physical dimensions of the garden, we need both functions to be positive:

1. For the length \( f(x) = x + 4 \):
- Set \( x + 4 > 0 \)
- \( x > -4 \)

2. For the width \( g(x) = 2x - 1 \):
- Set \( 2x - 1 > 0 \)
- \( 2x > 1 \)
- \( x > \frac{1}{2} \)

The more restrictive condition is \( x > \frac{1}{2} \). Therefore, the domain of the area function \( A(x) = 2x^2 + 7x - 4 \) is:
\[
\text{Domain: } x > \frac{1}{2}
\]

Finally, combining both parts, we can conclude:
- The function representing the area of the rectangular garden is \( A(x) = 2x^2 + 7x - 4 \).
- The domain is \( x > \frac{1}{2} \).