The area of a rectangular backyard is given by the function f(x)=x2−16

Question

. The length of a rectangular backyard is given by the function g(x)=x+4
. What function represents the width of the rectangular backyard? What is the domain?(1 point)
Responses

(fg)(x)=x2+x−12
and the domain of (fg)(x)
is x>4
.
open paren f over g close paren of x is equal to x squared plus x minus 12 and the domain of open paren f over g close paren of x is x is greater than 4 .

(fg)(x)=x2+x−12
and the domain of (fg)(x)
is (−∞,∞)
.
open paren f over g close paren of x is equal to x squared plus x minus 12 and the domain of open paren f over g close paren of x is open paren negative infinity comma infinity close paren .

(fg)(x)=x−4
and the domain of (fq)(x)
is x>−4
.
open paren f over g close paren of x is equal to x minus 4 and the domain of open paren f over q close paren times x is x is greater than negative 4 .

(fg)(x)=x−4
and the domain of (fg)(x)
is (−∞,∞)
.

Answer 1

To find the width of the rectangular backyard given the area function \( f(x) = x^2 - 16 \) and the length function \( g(x) = x + 4 \), we can express the width as:

\[ \text{Width} = \frac{\text{Area}}{\text{Length}} = \frac{f(x)}{g(x)} \]

Calculating this, we have:

\[ \text{Width} = \frac{x^2 - 16}{x + 4} \]

To simplify \( \frac{x^2 - 16}{x + 4} \), we can factor \( x^2 - 16 \) as \( (x - 4)(x + 4) \):

\[ \text{Width} = \frac{(x - 4)(x + 4)}{x + 4} \]

For \( x + 4 \neq 0 \), we can cancel \( x + 4 \):

\[ \text{Width} = x - 4 \quad \text{for } x \neq -4 \]

Thus, the function representing the width of the rectangular backyard is:

\[ \text{Width} = x - 4 \]

Domain

Since the width function \( x - 4 \) is derived from the cancelation of \( x + 4 \), we have to exclude the point where \( x + 4 = 0 \), which occurs at \( x = -4 \). Therefore, the domain of the width function is:

\[ x \in (-\infty, -4) \cup (-4, \infty) \]

However, since \( g(x) = x + 4 \) represents a length, the length must also be non-negative. This implies \( g(x) > 0 \), leading to:

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

Thus, to satisfy the physical conditions (length and width must be non-negative), the appropriate domain would be:

\[ x > -4 \]

Conclusion

Therefore, the function representing the width of the rectangular backyard is \( x - 4 \) and the domain is \( x > -4 \). The correct response would be:

(fg)(x) = x - 4 and the domain of (fg)(x) is x > -4.