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)=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
.

Answer 1

To find the function that represents the area of the rectangular garden, we need to multiply the functions \( f(x) \) and \( g(x) \):

1. Finding the functions:

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

2. Function for area: \[ A(x) = f(x) \cdot g(x) = (x + 4)(2x - 1) \]

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

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

4. Finding the domain: Since the length \( f(x) = x + 4 \) must be greater than zero for the garden to have a physical meaning (length cannot be negative), we solve:

\[ x + 4 > 0 \implies x > -4 \]

The width \( g(x) = 2x - 1 \) similarly must be positive:

\[ 2x - 1 > 0 \implies 2x > 1 \implies x > \frac{1}{2} \]

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

Thus, we can determine that the area function \( (f \cdot g)(x) = 2x^2 + 7x - 4 \) has the domain: \[ \text{Domain: } x > \frac{1}{2} \]

So, the complete result is: \[ (f \cdot g)(x) = 2x^2 + 7x - 4 \quad \text{and the domain of } (f \cdot g)(x) \text{ is } x > \frac{1}{2}. \]