Question

The length of a rectangle is 5 in. more than 3 times its width. The quadratic function f(x) = x(3x + 5) represents the area of the rectangle in terms of its width. What is a reasonable domain of the function?
A) real numbers
B) all real numbers less than 50
C) all real numbers less than zero
D) all real numbers greater than zero

Answer 1

Answer:

D) all real numbers greater than zero

Step-by-step explanation:

The area of the rectangle must be greater than zero. Because an area equal to zero or a negative area wouldn't make any sense. So let's write the inequalities in order to find the correct domain:

$f(x)=x(3x+5)>0$

Split in two equations:

$x>0\hspace{32}(1)\\\\3x+5>0\hspace{10}(2)$

From (1) we can determine directly that 5 must be greater than zero.

Now, for equation (2):

$3x+5>0\\\\3x>-5\\\\x>-\frac{5}{3}$

So, x can also take negative values less than -5/3, for example, let's evaluate the area for x=-2:

$f(-2)=-2(3(-2)+5)=-2(-6+5)=-2(-1)=2$

This would make sense for the area, however it wouldn't make sense for the side measurements. Never heard about of -10 ft for example.

Therefore the  domain of the function which make sense in this case is all real numbers greater than zero.

Answer 2

F(x)=x(3x+5)
or
f(x)=3x²+5x
Here the domain in general is: -∞<x<∞
But the length of the rectangle cannot be negative, therefore the domain is all real numbers greater than zero.

Option D is correct.