To determine the function that represents the width of the rectangular backyard, we need to relate the area \( f(x) \) and the length \( g(x) \) to find the width.
The area of a rectangle is given by:
\[ \text{Area} = \text{Length} \times \text{Width} \]
Given:
- Area function: \( f(x) = x^2 - 16 \)
- Length function: \( g(x) = x + 4 \)
Let \( w(x) \) represent the width. According to the area formula, we have:
\[ f(x) = g(x) \times w(x) \]
Substituting the definitions of \( f(x) \) and \( g(x) \):
\[ x^2 - 16 = (x + 4) \times w(x) \]
To find \( w(x) \), we can solve for it:
\[ w(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 16}{x + 4} \]
Now we can simplify \( \frac{x^2 - 16}{x + 4} \). Noticing that \( x^2 - 16 \) is a difference of squares:
\[ x^2 - 16 = (x - 4)(x + 4) \]
Therefore:
\[ w(x) = \frac{(x - 4)(x + 4)}{x + 4} \]
Since \( x + 4 \) is a common factor and we assume \( x + 4 \neq 0 \):
\[ w(x) = x - 4 \quad \text{for } x \neq -4 \]
As for the domain, the width function \( w(x) = x - 4 \) is defined for all \( x \) except \( x = -4 \), since the length \( g(x) \) cannot equal zero (which gives \( x + 4 = 0 \)).
Considering the length represents a physical dimension, the value must also be non-negative, which gives us:
\[ x + 4 > 0 \quad \Rightarrow \quad x > -4 \]
Thus, the relevant part of the domain is \( x > 4 \) (as the area also needs to be non-negative).
Therefore, the final answer is:
- The function representing the width is \( w(x) = x - 4 \),
- The domain is \( x > 4 \).
Thus, the second response is correct: \( (fg)(x) = x - 4 \) and the domain of \( (fg)(x) \) is \( x > 4 \).
1 answer