Question

Erin buys gas at a self service station for $2.75 a gallon. The gas station has a promotion going on that anyone who buys more than 10 gallons of gas, only has to pay $2.50 per gallon. Erin's tank will hold 12 gallons of gas.
a) Write a rule for the total cost, C(g), as a function of g gallons of gas.
b)Graph the piecewise function
c)What is the domain and range of the function in part a

Answer 1

@jim_thompson5910 help please

Answer 2

what do you have so far?

Answer 3

@jim_thompson5910 not much sir, i'm confused actually

Answer 4

`Erin buys gas at a self service station for $2.75 a gallon`

if Erin buys g gallons of gas, then how much does she spend?

Answer 5

how many gallons does Erin buy though?

Answer 6

ok let's say she buys 5 gallons

how much does she spend?

Answer 7

it'll be $13.75 then

Answer 8

how did you compute that

Answer 9

by multiplying the number of gallons Erin was going to buy with the price that each gallon costs

Answer 10

so in general, we can say `2.75*g` or just `2.75g`

Answer 11

g is it the unknown number of gallons bought

Answer 12

okay yeah i understand that

Answer 13

If the promotion wasn't there, then C(g) = 2.75g

and that would be it. There wouldn't be a need for a piecewise function

Answer 14

however, the promotion changes things

C(g) = 2.75g is only true if \(\Large 0 \le g \le 10\)

anything larger than 10 will have a different price structure

Answer 15

so it wouldn't be just greater than? it's greater than or equal to right

Answer 16

so that'll be the rule for letter a?

Answer 17

not the full rule

Answer 18

oh okay so then what else do we need?

Answer 19

`The gas station has a promotion going on that anyone who buys more than 10 gallons of gas, only has to pay $2.50 per gallon.`

what is the cost function if you buy more than 10 gallons of gas?

Answer 20

i seriously dont know D:

Answer 21

same as before

total cost = (price per gallon)*(number of gallons)

keep the number of gallons as g
the price will change

Answer 22

C(g)=2.50g

Answer 23

what are the restrictions on g?

Answer 24

do you pay $2.50 per gallon if you bought say, 8 gallons?

Answer 25

no, it's when you buy more than 10 gallons so wouldn't it be \[0\le g \le 10\]

Answer 26

more like g > 10

if g > 10, then the cost function is C(g) = 2.50g

Answer 27

now let's form the piecewise function

Answer 28

Everything in blue
\[\Large \color{blue}{
\Large C(g) = 2.75g \text{ if } 0 \le g \le 10 \\
\Large \text{OR}\\
\Large C(g) = 2.50g \text{ if } g > 10 \\
}\]
condenses to this piecewise function notation you see in red
\[\Large \color{red}{C(g)=\begin{cases}2.75g \text{ if } 0 \le g \le 10 \\ 2.50g \text{ if } g > 10 \end{cases}}\]
The stuff in red is what most math books will use as piecewise function notation. The stuff in blue is more intuitive (at least to me), so it's good to know both forms.

Answer 29

take a few moments to write down notes and process all this. I want to make sure you understand all this before we move on

Answer 30

ah okay i see what you did there, so basically what's in blue is the rule for the total cost correct?

Answer 31

and then what's in red is basically the rule for the total cost but more "like what textbooks would use" right

Answer 32

yeah the cost function is either one or the other (pick one) and it's based on what g is

if g is between 0 and 10, you pick the first one
if g is greater than 10, you pick the second one

Answer 33

correct, it's the formal mathematical way

Answer 34

ah okay this is making so much more sense now, okay im done taking notes now

Answer 35

once you get to the red stuff, you are done with part a)

Answer 36

ready for part b)?

Answer 37

yeah

Answer 38

to graph a piecewise function, we simply graph each piece on the same xy axis

so focus on just the first piece C(g) = 2.75g

do you know how to graph that?

Answer 39

tbh no i dont how will you start it?

Answer 40

think of it as y = 2.75x

Answer 41

what kind of equation is that?

Answer 42

hint: y = mx+b

Answer 43

it's slope intercept form right?

Answer 44

yeah we have slope m = 2.75 and y intercept b = 0

do you see how I'm getting that?

Answer 45

yeah i see how plus i just graphed it on my calculator and it gave me the coordinates to graph.

Answer 46

what two points are on this graph

Answer 47

two points are (1,2.75) and (2,5.5)

Answer 48

good

Answer 49

and then the second one will be y=2.50x ?

Answer 50

so normally, when you graph y = 2.75x, you draw a straight line through those two points

Answer 51

however, there's a restriction on x

in this case, \(\Large 0 \le x \le 10\) is the restriction

Answer 52

so does that mean stop graphing when the x coordinate is at 10?

Answer 53

so what that means is that you only graph the portion between x = 0 and x = 10

you erase everything else

Answer 54

make sense?

Answer 55

yeah it makes sense

Answer 56

and how will that be different for c(g)=2.50g

Answer 57

now we plot closed circles at the endpoints

because we are including the endpoints