function equations

Question

A car rental company charged a flat fee of $30 for rides 15 miles or less. For 16-30 mile trips, the

charge is $1.20 per mile for every mile with no flat fee. The charge for a ride greater than 30 miles  is a $1.10 per mile with no flat rate. 

Please write a function for the above problem and then use it to plot these points ƒ(12), ƒ(23), ƒ(37)

 ƒ(x)=$51.70, f(x) = $28.80 on the the graph below.

Answer 1

Interesting...

Think of the FLAT FEE as the initial value of y when x = 0. This point will be (0, 30) and the function for when the mileage is 0 - 15 will be: y OR f(x) = 30.

For any value of x (up to 15), y = 30. This is a horizontal line on the graph.

When the mileage ranges from 16 - 30, there is NO flat fee and now y or f(x) is dependent on the value of x (miles) and the rate (or slope) of $1.10/mile, The company simply charges for the miles driven.

y OR f(x) = 1.10x

When the mileage is > 30, there is NO flat fee and now y or f(x) is dependent on the value of x (miles) and the rate (or slope) of $1.20/mile, so:

y OR f(x) = 1.20x

So f(12) is the value of y when x = 12. Since 12 is btw 0 - 15, the value of y or f(x) will meet condition #1 (mileage btw 0 and 15), where only the FLAT FEE is charged

f(12) = 30

The ordered point will be (12, 30)

For f(23), the value of x is now btw (16 - 30) meeting the 2nd condition. So you need to compute the value of y or f(x) when x = 23 using the 2nd function f(x) = 1.10x

f(23) = 1.10x = 1.10(23) = 25.3

The ordered point will be (23, 25.3)

Same as above, f(37) = 1.20x (because it meets the 3rd condition x or the mileage is > 30)

f(37) = 1.20x = 1.20(37) = 44.4

The ordered point will be (37, 44.4)

For the last part of the problem, you are given the value of y or f(x), so you have to determine the corresponding x value.

f(x) = $51.70 and f(x) $28.80

From the previous calculations, I think its safe to assume that with f(x) = 51.70 the value of x will be greater than 30 (meeting the 3rd condition) since at x = 37, y was equal to 44.4.

f(x) = 1.20x

51.70 = 1.20x

Solve for x

x = 51.70/1.20 = 43.08 or 43.1

So this ordered point will be (43.1, 51.7)

f(x) = 28.80, it is safe to assume that the value of x will fall btw 16 and 30, because its value is very clost to the value of f(x) when x was equal to 23, meeting the 2nd condition

f(x) = 1.10x

28.80 = 1.10x

Solve for x

x = 28.80/1.10 = 26.1

So this ordered point will be (26.1, 28.8)

Hope this helps :)

Answer 2

f(x) = 30 for x<15

= 1.2M for 15<M<30

= 1.1M for M>30

It's a step function, graphically, a flat line with slope =0 =, then a straight line with slope=1.2, and then another straight line with slope 1.1

f(12) =30, f(23) =1.2(23) = 49.6, f(37)=1.1(37)=40.7

f(x)=51.7 with x = 51.7/1.1 = 47

f(x)=28.8 with x=28.8/1.2 = 24