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.

Explanation:

Step1: Expand the function

\[

$$\begin{align*} f(x)&=\frac{1}{2}x(14 - 2x)\\ &=\frac{1}{2}(14x-2x^{2})\\ &=7x - x^{2} \end{align*}$$

\]

Step2: Find function - values

Let's assume we want to find \(f(1)\):
\[

$$\begin{align*} f(1)&=7\times1-1^{2}\\ &=7 - 1\\ &=6 \end{align*}$$

\]
Interpretation: When the width of the rectangle is \(1\) foot, the area of the fenced - in rectangle is \(6\) square feet.

Step3: Determine the domain

The width \(x>0\) and the length \(\frac{1}{2}(14 - 2x)>0\).
Solve \(\frac{1}{2}(14 - 2x)>0\):
\[

$$\begin{align*} 14-2x&>0\\ -2x&>- 14\\ x&<7 \end{align*}$$

\]
So the domain is \(0 < x<7\).

Answer:

Function values can be found by substituting values of \(x\) into \(f(x)=7x - x^{2}\). For example, \(f(1) = 6\) means when the width is \(1\) foot, the area is \(6\) square feet. The domain of the function \(f(x)\) is \(0 < x<7\).