Answer:
Interval for constant cost = [0, 10,000] same as 0 <= x <= 10,000
Interval for increasing cost = (10,000, ∞) same as 10,000 < x < ∞
Be careful! note the use of different types of parentheses for each answer part as explained below
Explanation:
We are only being asked for the two intervals, not the function definition itself which would have made it a little bit more work. A function has inputs and outputs. Here the input is gallons of water and output is $ cost.
Normally when asked for intervals we use a variable. Let's call it x which represents the gallons of water used
Since there is a base rate of $11.70 whether or not any water is consumed as long as it is less than or equal to 10,000 gallons we can state that over the interval 0 to 10000 (inclusive), the cost is constant
We can represent this interval in two ways
0 ≤ x ≤ 10,000
Another way of representing this interval is what we call interval notation. In interval notation the same thing can be expressed as:
[0, 10000]
Here 0 is referred to as the lower limit or bound(minimum) and the 10,000 is referred to as the upper limit or upper bound
Note the use of square brackets. This means that both upper and lower limits are included in the function. Such an interval is also called a closed interval.
After 10,000 gallons, the cost keeps increasing. There is no upper limit on the usage of water. So the lower limit is 10,000 (but does not include that number) and upper limit is infinity
We represent this interval as
10000 < x < ∞
Note that both upper and lower limits are excluded since ∞ is not realistic.
In interval notation this is represented as
(10000, ∞)
Here we are using regular brackets(parentheses) because this interval does not include lower and upper limits
This kind of interval is known as an open interval
Looks like the purpose of this question is to make sure what you understand in terms of interval notation and possibly open and closed intervals. The only relevant number there is the 10,000 gallons which represents the upper bound on the first interval. All the other numbers can be junked
Hope that is of help even if it was wordy