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

- ineedhelp10

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- ineedhelp10

@jim_thompson5910 help please

- jim_thompson5910

what do you have so far?

- ineedhelp10

@jim_thompson5910 not much sir, i'm confused actually

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## More answers

- jim_thompson5910

`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?

- ineedhelp10

how many gallons does Erin buy though?

- jim_thompson5910

ok let's say she buys 5 gallons
how much does she spend?

- ineedhelp10

it'll be $13.75 then

- jim_thompson5910

how did you compute that

- ineedhelp10

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

- jim_thompson5910

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

- jim_thompson5910

g is it the unknown number of gallons bought

- ineedhelp10

okay yeah i understand that

- jim_thompson5910

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

- jim_thompson5910

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

- ineedhelp10

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

- ineedhelp10

so that'll be the rule for letter a?

- jim_thompson5910

not the full rule

- ineedhelp10

oh okay so then what else do we need?

- jim_thompson5910

`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?

- ineedhelp10

i seriously dont know D:

- jim_thompson5910

same as before
total cost = (price per gallon)*(number of gallons)
keep the number of gallons as g
the price will change

- ineedhelp10

C(g)=2.50g

- jim_thompson5910

what are the restrictions on g?

- jim_thompson5910

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

- ineedhelp10

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

- jim_thompson5910

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

- jim_thompson5910

now let's form the piecewise function

- jim_thompson5910

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.

- jim_thompson5910

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

- ineedhelp10

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

- ineedhelp10

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

- jim_thompson5910

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

- jim_thompson5910

correct, it's the formal mathematical way

- ineedhelp10

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

- jim_thompson5910

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

- jim_thompson5910

ready for part b)?

- ineedhelp10

yeah

- jim_thompson5910

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?

- ineedhelp10

tbh no i dont how will you start it?

- jim_thompson5910

think of it as y = 2.75x

- jim_thompson5910

what kind of equation is that?

- jim_thompson5910

hint: y = mx+b

- ineedhelp10

it's slope intercept form right?

- jim_thompson5910

yeah we have slope m = 2.75 and y intercept b = 0
do you see how I'm getting that?

- ineedhelp10

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

- jim_thompson5910

what two points are on this graph

- ineedhelp10

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

- jim_thompson5910

good

- ineedhelp10

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

- jim_thompson5910

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

- jim_thompson5910

however, there's a restriction on x
in this case, \(\Large 0 \le x \le 10\) is the restriction

- ineedhelp10

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

- jim_thompson5910

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

- jim_thompson5910

|dw:1442368336119:dw|

- jim_thompson5910

|dw:1442368346952:dw|

- jim_thompson5910

|dw:1442368378189:dw|

- jim_thompson5910

make sense?

- ineedhelp10

yeah it makes sense

- ineedhelp10

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

- jim_thompson5910

now we plot closed circles at the endpoints
because we are including the endpoints
|dw:1442368517079:dw|

- jim_thompson5910

then we move onto graphing y = 2.50x
this is only graphed when x > 10

- ineedhelp10

so how will that one look? that one looks more confusing

- jim_thompson5910

so graph y = 2.50x like normal
then erase the portion to the left of x = 10

- ineedhelp10

wait you think you can draw an example of it please?

- jim_thompson5910

what two points are on y = 2.50x

- ineedhelp10

(1,2.5) and (2.5)

- jim_thompson5910

(2,5) you mean?

- ineedhelp10

yeah lol sorry that's what i meant

- jim_thompson5910

plot the two points, draw a line through them
that will get you the graph of y = 2.50x

- jim_thompson5910

then erase the portion to the left of x = 10

- ineedhelp10

that's whats confusing me when you say left of x=10 like where on the graph is that ?

- jim_thompson5910

|dw:1442368737285:dw|

- jim_thompson5910

|dw:1442368751023:dw|

- jim_thompson5910

|dw:1442368771771:dw|

- jim_thompson5910

|dw:1442368802811:dw|

- ineedhelp10

ah okay so then the first point wouldn't be (10,25) right?

- jim_thompson5910

you'll have this left over
|dw:1442368832183:dw|

- jim_thompson5910

yes

- jim_thompson5910

there is an open circle at (10,25) because it is NOT included
this is because there is no line under the `>` in `g > 10`
|dw:1442368915013:dw|

- ineedhelp10

ah okay i get the graphing now

- jim_thompson5910

let's put the two pieces together to form one single graph
|dw:1442368938829:dw|

- jim_thompson5910

here's what desmos shows
https://www.desmos.com/calculator/oyvslmmg3f
so it confirms things. The only thing missing is the open & closed circles at the endpoints

- ineedhelp10

ah okay sounds good so far

- ineedhelp10

what about the domain and range of the function in part a?

- jim_thompson5910

tell me what you think the domain would be

- ineedhelp10

will the domain be anything greater than or equal to 10? :///

- ineedhelp10

no wait would it be 12 since that's how many gallons erin can hold in his car?

- jim_thompson5910

you're forgetting the stuff lower than 10

- ineedhelp10

can you explain the domain please

- jim_thompson5910

you're right, she can only hold 12 gallons of gas
so the domain would be the set of numbers between 0 and 12
the domain is the set of allowed numbers of gallons of gas that can be bought (in this specific case, just for Erin)

- ineedhelp10

so it'll be \[x \le 12\]

- jim_thompson5910

actually
\[\Large 0 \le g \le 12\]

- ineedhelp10

and what about the range? i always had trouble with range

- ineedhelp10

@jim_thompson5910 are you there still?

- jim_thompson5910

sorry I went away from my computer for a moment

- jim_thompson5910

do you see how I got that domain?

- ineedhelp10

yeah i saw how now

- ineedhelp10

im just confused on the range now

- jim_thompson5910

what is the smallest value in the domain?

- ineedhelp10

0

- jim_thompson5910

plug that into C(g) to get ??

- ineedhelp10

0 again

- jim_thompson5910

so 0 is the smallest number in the range

- jim_thompson5910

what is the largest number in the domain?

- ineedhelp10

its 12

- jim_thompson5910

if g = 12, then C(g) = ??

- ineedhelp10

$30 ?

- jim_thompson5910

so the range spans from 0 to 30 (including both endpoints)

- ineedhelp10

how will that look like when writing it down?

- jim_thompson5910

well you can either say
\[\Large 0 \le y \le 30\]
or
\[\Large 0 \le C(g) \le 30\]

- ineedhelp10

ah okay i get it now thanks soooooooooo much!! i really appreciate you taking time to help me out on this problem i appreciate it!

- jim_thompson5910

I'm glad to be of help

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