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

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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

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@jim_thompson5910 help please
what do you have so far?
@jim_thompson5910 not much sir, i'm confused actually

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`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?
how many gallons does Erin buy though?
ok let's say she buys 5 gallons how much does she spend?
it'll be $13.75 then
how did you compute that
by multiplying the number of gallons Erin was going to buy with the price that each gallon costs
so in general, we can say `2.75*g` or just `2.75g`
g is it the unknown number of gallons bought
okay yeah i understand that
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
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
so it wouldn't be just greater than? it's greater than or equal to right
so that'll be the rule for letter a?
not the full rule
oh okay so then what else do we need?
`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?
i seriously dont know D:
same as before total cost = (price per gallon)*(number of gallons) keep the number of gallons as g the price will change
C(g)=2.50g
what are the restrictions on g?
do you pay $2.50 per gallon if you bought say, 8 gallons?
no, it's when you buy more than 10 gallons so wouldn't it be \[0\le g \le 10\]
more like g > 10 if g > 10, then the cost function is C(g) = 2.50g
now let's form the piecewise function
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.
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
ah okay i see what you did there, so basically what's in blue is the rule for the total cost correct?
and then what's in red is basically the rule for the total cost but more "like what textbooks would use" right
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
correct, it's the formal mathematical way
ah okay this is making so much more sense now, okay im done taking notes now
once you get to the red stuff, you are done with part a)
ready for part b)?
yeah
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?
tbh no i dont how will you start it?
think of it as y = 2.75x
what kind of equation is that?
hint: y = mx+b
it's slope intercept form right?
yeah we have slope m = 2.75 and y intercept b = 0 do you see how I'm getting that?
yeah i see how plus i just graphed it on my calculator and it gave me the coordinates to graph.
what two points are on this graph
two points are (1,2.75) and (2,5.5)
good
and then the second one will be y=2.50x ?
so normally, when you graph y = 2.75x, you draw a straight line through those two points
however, there's a restriction on x in this case, \(\Large 0 \le x \le 10\) is the restriction
so does that mean stop graphing when the x coordinate is at 10?
so what that means is that you only graph the portion between x = 0 and x = 10 you erase everything else
|dw:1442368336119:dw|
|dw:1442368346952:dw|
|dw:1442368378189:dw|
make sense?
yeah it makes sense
and how will that be different for c(g)=2.50g
now we plot closed circles at the endpoints because we are including the endpoints |dw:1442368517079:dw|
then we move onto graphing y = 2.50x this is only graphed when x > 10
so how will that one look? that one looks more confusing
so graph y = 2.50x like normal then erase the portion to the left of x = 10
wait you think you can draw an example of it please?
what two points are on y = 2.50x
(1,2.5) and (2.5)
(2,5) you mean?
yeah lol sorry that's what i meant
plot the two points, draw a line through them that will get you the graph of y = 2.50x
then erase the portion to the left of x = 10
that's whats confusing me when you say left of x=10 like where on the graph is that ?
|dw:1442368737285:dw|
|dw:1442368751023:dw|
|dw:1442368771771:dw|
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ah okay so then the first point wouldn't be (10,25) right?
you'll have this left over |dw:1442368832183:dw|
yes
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|
ah okay i get the graphing now
let's put the two pieces together to form one single graph |dw:1442368938829:dw|
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
ah okay sounds good so far
what about the domain and range of the function in part a?
tell me what you think the domain would be
will the domain be anything greater than or equal to 10? :///
no wait would it be 12 since that's how many gallons erin can hold in his car?
you're forgetting the stuff lower than 10
can you explain the domain please
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)
so it'll be \[x \le 12\]
actually \[\Large 0 \le g \le 12\]
and what about the range? i always had trouble with range
@jim_thompson5910 are you there still?
sorry I went away from my computer for a moment
do you see how I got that domain?
yeah i saw how now
im just confused on the range now
what is the smallest value in the domain?
0
plug that into C(g) to get ??
0 again
so 0 is the smallest number in the range
what is the largest number in the domain?
its 12
if g = 12, then C(g) = ??
$30 ?
so the range spans from 0 to 30 (including both endpoints)
how will that look like when writing it down?
well you can either say \[\Large 0 \le y \le 30\] or \[\Large 0 \le C(g) \le 30\]
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!
I'm glad to be of help

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