A taxi company charges $3.00 for the first mile (or part of a mile) and 60 cents for each succeeding tenth of a mile (or part). Express the cost C (in dollars) of a ride as a piecewise defined function of the distance x traveled (in miles) for 0 < x ≤ 2.

Time to review step functions in your text.

You know that [x] is the greatest integer less than or equal to x.
As a segment on the number line, each step looks like
•---------◦
But we want our steps to look like
◦---------•
That is, as soon as we travel more than, say, 0.3 miles, we start charging for 0.4 miles.
Now, note that for positive x, [x] looks like the first diagram, but for negative x, it looks like the second one. This is because [-4.3] = -5, not -4. That is, -5 is the greatest integer less than -4.3

Now we can see how to build our taxi function.

C(x) = 3.00 if 0 < x <= 1
C(x) = 3.00 - 0.60[-10(x-1)] if 1 < x <= 2

e

Why did the taxi driver become a mathematician? Because he loved calculating fares and turning a profit! Now let's solve this math problem together.

For the first mile (or part of a mile), the cost is $3.00. So if 0 < x ≤ 1, the cost is $3.00.

For each succeeding tenth of a mile (or part), the cost is 60 cents. So if 1 < x ≤ 2, we need to calculate the number of tenths of a mile beyond the first mile. Since there are 10 tenths in a mile, we can use x - 1 to find how many tenths beyond the first mile.

Therefore, the cost C (in dollars) of a ride can be expressed as a piecewise defined function:

C(x) =
- 3.00, if 0 < x ≤ 1
- 3.00 + 0.60(x - 1), if 1 < x ≤ 2

Remember, this function calculates the cost of a ride when 0 < x ≤ 2, considering the initial $3.00 charge and the additional 60 cents for each succeeding tenth of a mile. Happy calculating and safe travels!

To express the cost of a taxi ride as a piecewise defined function, we need to consider the different intervals of the distance traveled.

For distances 0 < x ≤ 1, the cost is always $3.00 since it includes the first mile or part of a mile. So, we can define this interval as:

C(x) = 3, for 0 < x ≤ 1

For distances 1 < x ≤ 2, the cost includes the $3.00 for the first mile or part, and then an additional cost for each succeeding tenth of a mile. In this case, the cost increases by 60 cents for each tenth of a mile. So, we need to calculate the extra cost based on the distance beyond the first mile. Let's define this interval as:

C(x) = 3 + 0.60 * (x - 1), for 1 < x ≤ 2

This means that the cost for distances between 1 and 2 miles will be the initial $3.00 plus 60 cents for each tenth of a mile beyond the first mile.

In summary, the cost of a ride as a piecewise function of the distance traveled for 0 < x ≤ 2 is:

C(x) = 3, for 0 < x ≤ 1
C(x) = 3 + 0.60 * (x - 1), for 1 < x ≤ 2