Question

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.

Answer 1

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

Answer 2

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