To find the width of the rectangular backyard, we need to relate the area function \( f(x) \) and the length function \( g(x) \) through the formula for the area of a rectangle, which is:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Area function: \( f(x) = x^2 - 16 \)
- Length function: \( g(x) = x + 4 \)
We denote the width as \( w(x) \). According to the area formula:
\[ f(x) = g(x) \cdot w(x) \]
Substituting the known functions:
\[ x^2 - 16 = (x + 4) \cdot w(x) \]
To solve for \( w(x) \), we can divide both sides by \( g(x) = x + 4 \):
\[ w(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 16}{x + 4} \]
Next, we can simplify \( \frac{x^2 - 16}{x + 4} \). The expression \( x^2 - 16 \) can be factored:
\[ x^2 - 16 = (x - 4)(x + 4) \]
Thus,
\[ w(x) = \frac{(x - 4)(x + 4)}{x + 4} \]
When we cancel out \( x + 4 \) (noting that we cannot allow \( x + 4 = 0 \) in order to avoid division by zero), we have:
\[ w(x) = x - 4 \quad \text{for } x \neq -4 \]
Now, regarding the domain, we need to consider the original function \( g(x) = x + 4 \), which cannot equal zero. Therefore:
\[ x + 4 \neq 0 \implies x \neq -4 \]
The domain for \( w(x) \) will be all real numbers except \( -4 \):
\[ \text{Domain: } (-\infty, -4) \cup (-4, \infty) \]
The final result is:
\[ w(x) = x - 4 \quad \text{with domain } (-\infty, -4) \cup (-4, \infty) \]
Among the provided responses, none accurately represent this conclusion. The closest option concerning the function \( w(x) \) being \( x - 4 \) and the domain not being accurate seems to be presented ambiguously.
However, from the mathematical process, it's clear the width function is \( w(x) = x - 4 \), excluding the point where \( x = -4 \) in its domain.