Question

A bucket has 16 inches of water in it but there is a hole in the bottom. Use interval notation to write the domain and range of the function H(x) = 16 - 1.25x, where x is the time the water has been leaking for (in minutes) and is the height of the water (in inches). *

Answer 1

Answer: range:  {0in, 16in}

               domain {0min, 12.8 min}

Step-by-step explanation:

When we have a function:

f(x) = y.

The range is the set of possible values of y and the domain is the set of all the possible values of x.

In this case, our function is:

H(x) = 16 - 1.25*x

which is a linear equation, and we know that the linear equations are defined (for range and domain) in the set of all the real numbers, but this is a physical situation, so we must see at the real problem.

The bucket can not have more water than the initial amount, 16 inches, so this is the maximum in the range.

The minimum height of water that we can find in the bucket is 0 inches (so the bucket is empty) then this is the minimum of the range.

Then we can write the range as:

R: 0in ≤ y ≤ 16in. = {0in, 16in}

Now we can find the extremes of the domain by using the extremes of the range:

y = 16 = 16 - 1.25*x

       0 = -1.25*x

then we have x = 0min, this will be the minimum of the domain.

Now using the minimum of the range y = 0 we have:

y  = 0 = 16 - 1.25*x

      1.25*x = 16

            x  = 16/1.25 = 12.8 mins

This is the maximum time in the domain (because after this time, there is no water in the bucket)

Then the domain is:

D: 0min ≤ x ≤ 12.8 min