Question

taylor has 14 feet of fence available to build a rectangular fenced - in area. if the width of the rectangle is x feet, then the length would be $\frac{1}{2}(14 - 2x)$. a function to find the area, in square feet, of the fenced - in rectangle with width x is given by $f(x)=\frac{1}{2}x(14 - 2x)$. find and interpret the given function values and determine an appropriate domain for the function.
answer attempt 1 out of 2
$f(-4)= square$, meaning when the width of the rectangular area is $square$ ft, the area would be $square$ $ft^{2}$. this interpretation in the context of the problem.

Explanation:

Step1: Substitute x = - 4 into the function

$f(-4)=\frac{1}{2}\times(-4)\times(14 - 2\times(-4))$

Step2: Simplify the expression inside the parentheses

$14-2\times(-4)=14 + 8=22$

Step3: Calculate the product

$f(-4)=\frac{1}{2}\times(-4)\times22=-44$

The width of a rectangle cannot be negative in the real - world context of fencing a rectangular area.

For the domain, since the width $x>0$ and the length $\frac{1}{2}(14 - 2x)>0$. Solving $\frac{1}{2}(14 - 2x)>0$ gives $14-2x>0$, then $2x<14$, $x < 7$.

Answer:

$f(-4)=-44$, meaning when the width of the rectangular area is $- 4$ ft, the area would be $-44$ $ft^{2}$. This interpretation is not valid in the context of the problem. The domain of the function is $0