To find the area of the base of the box and the domain, we need to analyze the given functions:
-
Volume of the box: The volume \( V \) is represented by the function: \[ f(x) = x^3 - 8 \]
-
Height of the box: The height \( h \) is given by the function: \[ g(x) = x - 2 \]
The volume \( V \) of a box (rectangular prism) is calculated by multiplying the area of the base \( A \) by the height \( h \): \[ V = A \times h \]
From this, we can express the area of the base \( A \) as: \[ A = \frac{V}{h} = \frac{f(x)}{g(x)} = \frac{x^3 - 8}{x - 2} \]
To simplify \( A = \frac{x^3 - 8}{x - 2} \), we can perform polynomial long division or factor the numerator. \( x^3 - 8 \) can be factored as: \[ x^3 - 8 = (x - 2)(x^2 + 2x + 4) \]
Thus, \[ A = \frac{(x - 2)(x^2 + 2x + 4)}{x - 2} \]
Assuming \( x - 2 \neq 0 \) (i.e., \( x \neq 2 \)), we can cancel the \( x - 2 \) terms: \[ A = x^2 + 2x + 4 \]
Area and Domain:
-
The area of the base of the box is given by: \[ A = x^2 + 2x + 4 \]
-
The domain of \( A \) is restricted by the condition \( g(x) = x - 2 \neq 0 \): \[ x - 2 > 0 \implies x > 2 \]
Thus, the area of the base of the box is \( x^2 + 2x + 4 \), and the domain is \( x > 2 \).
So the correct response based on your options appears to be:
The function \( (fg)(x) = x^2 + 2x + 4 \) represents the area of the base of the box. The domain is \( x > 2 \).