ineedhelp10
  • ineedhelp10
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
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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ineedhelp10
  • ineedhelp10
@jim_thompson5910 help please
jim_thompson5910
  • jim_thompson5910
what do you have so far?
ineedhelp10
  • ineedhelp10
@jim_thompson5910 not much sir, i'm confused actually

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jim_thompson5910
  • 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
  • ineedhelp10
how many gallons does Erin buy though?
jim_thompson5910
  • jim_thompson5910
ok let's say she buys 5 gallons how much does she spend?
ineedhelp10
  • ineedhelp10
it'll be $13.75 then
jim_thompson5910
  • jim_thompson5910
how did you compute that
ineedhelp10
  • ineedhelp10
by multiplying the number of gallons Erin was going to buy with the price that each gallon costs
jim_thompson5910
  • jim_thompson5910
so in general, we can say `2.75*g` or just `2.75g`
jim_thompson5910
  • jim_thompson5910
g is it the unknown number of gallons bought
ineedhelp10
  • ineedhelp10
okay yeah i understand that
jim_thompson5910
  • 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
  • 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
  • ineedhelp10
so it wouldn't be just greater than? it's greater than or equal to right
ineedhelp10
  • ineedhelp10
so that'll be the rule for letter a?
jim_thompson5910
  • jim_thompson5910
not the full rule
ineedhelp10
  • ineedhelp10
oh okay so then what else do we need?
jim_thompson5910
  • 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
  • ineedhelp10
i seriously dont know D:
jim_thompson5910
  • 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
  • ineedhelp10
C(g)=2.50g
jim_thompson5910
  • jim_thompson5910
what are the restrictions on g?
jim_thompson5910
  • jim_thompson5910
do you pay $2.50 per gallon if you bought say, 8 gallons?
ineedhelp10
  • ineedhelp10
no, it's when you buy more than 10 gallons so wouldn't it be \[0\le g \le 10\]
jim_thompson5910
  • jim_thompson5910
more like g > 10 if g > 10, then the cost function is C(g) = 2.50g
jim_thompson5910
  • jim_thompson5910
now let's form the piecewise function
jim_thompson5910
  • 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
  • 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
  • ineedhelp10
ah okay i see what you did there, so basically what's in blue is the rule for the total cost correct?
ineedhelp10
  • 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
  • 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
  • jim_thompson5910
correct, it's the formal mathematical way
ineedhelp10
  • ineedhelp10
ah okay this is making so much more sense now, okay im done taking notes now
jim_thompson5910
  • jim_thompson5910
once you get to the red stuff, you are done with part a)
jim_thompson5910
  • jim_thompson5910
ready for part b)?
ineedhelp10
  • ineedhelp10
yeah
jim_thompson5910
  • 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
  • ineedhelp10
tbh no i dont how will you start it?
jim_thompson5910
  • jim_thompson5910
think of it as y = 2.75x
jim_thompson5910
  • jim_thompson5910
what kind of equation is that?
jim_thompson5910
  • jim_thompson5910
hint: y = mx+b
ineedhelp10
  • ineedhelp10
it's slope intercept form right?
jim_thompson5910
  • jim_thompson5910
yeah we have slope m = 2.75 and y intercept b = 0 do you see how I'm getting that?
ineedhelp10
  • ineedhelp10
yeah i see how plus i just graphed it on my calculator and it gave me the coordinates to graph.
jim_thompson5910
  • jim_thompson5910
what two points are on this graph
ineedhelp10
  • ineedhelp10
two points are (1,2.75) and (2,5.5)
jim_thompson5910
  • jim_thompson5910
good
ineedhelp10
  • ineedhelp10
and then the second one will be y=2.50x ?
jim_thompson5910
  • jim_thompson5910
so normally, when you graph y = 2.75x, you draw a straight line through those two points
jim_thompson5910
  • jim_thompson5910
however, there's a restriction on x in this case, \(\Large 0 \le x \le 10\) is the restriction
ineedhelp10
  • ineedhelp10
so does that mean stop graphing when the x coordinate is at 10?
jim_thompson5910
  • 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
  • jim_thompson5910
|dw:1442368336119:dw|
jim_thompson5910
  • jim_thompson5910
|dw:1442368346952:dw|
jim_thompson5910
  • jim_thompson5910
|dw:1442368378189:dw|
jim_thompson5910
  • jim_thompson5910
make sense?
ineedhelp10
  • ineedhelp10
yeah it makes sense
ineedhelp10
  • ineedhelp10
and how will that be different for c(g)=2.50g
jim_thompson5910
  • jim_thompson5910
now we plot closed circles at the endpoints because we are including the endpoints |dw:1442368517079:dw|
jim_thompson5910
  • jim_thompson5910
then we move onto graphing y = 2.50x this is only graphed when x > 10
ineedhelp10
  • ineedhelp10
so how will that one look? that one looks more confusing
jim_thompson5910
  • jim_thompson5910
so graph y = 2.50x like normal then erase the portion to the left of x = 10
ineedhelp10
  • ineedhelp10
wait you think you can draw an example of it please?
jim_thompson5910
  • jim_thompson5910
what two points are on y = 2.50x
ineedhelp10
  • ineedhelp10
(1,2.5) and (2.5)
jim_thompson5910
  • jim_thompson5910
(2,5) you mean?
ineedhelp10
  • ineedhelp10
yeah lol sorry that's what i meant
jim_thompson5910
  • jim_thompson5910
plot the two points, draw a line through them that will get you the graph of y = 2.50x
jim_thompson5910
  • jim_thompson5910
then erase the portion to the left of x = 10
ineedhelp10
  • ineedhelp10
that's whats confusing me when you say left of x=10 like where on the graph is that ?
jim_thompson5910
  • jim_thompson5910
|dw:1442368737285:dw|
jim_thompson5910
  • jim_thompson5910
|dw:1442368751023:dw|
jim_thompson5910
  • jim_thompson5910
|dw:1442368771771:dw|
jim_thompson5910
  • jim_thompson5910
|dw:1442368802811:dw|
ineedhelp10
  • ineedhelp10
ah okay so then the first point wouldn't be (10,25) right?
jim_thompson5910
  • jim_thompson5910
you'll have this left over |dw:1442368832183:dw|
jim_thompson5910
  • jim_thompson5910
yes
jim_thompson5910
  • 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
  • ineedhelp10
ah okay i get the graphing now
jim_thompson5910
  • jim_thompson5910
let's put the two pieces together to form one single graph |dw:1442368938829:dw|
jim_thompson5910
  • 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
  • ineedhelp10
ah okay sounds good so far
ineedhelp10
  • ineedhelp10
what about the domain and range of the function in part a?
jim_thompson5910
  • jim_thompson5910
tell me what you think the domain would be
ineedhelp10
  • ineedhelp10
will the domain be anything greater than or equal to 10? :///
ineedhelp10
  • ineedhelp10
no wait would it be 12 since that's how many gallons erin can hold in his car?
jim_thompson5910
  • jim_thompson5910
you're forgetting the stuff lower than 10
ineedhelp10
  • ineedhelp10
can you explain the domain please
jim_thompson5910
  • 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
  • ineedhelp10
so it'll be \[x \le 12\]
jim_thompson5910
  • jim_thompson5910
actually \[\Large 0 \le g \le 12\]
ineedhelp10
  • ineedhelp10
and what about the range? i always had trouble with range
ineedhelp10
  • ineedhelp10
@jim_thompson5910 are you there still?
jim_thompson5910
  • jim_thompson5910
sorry I went away from my computer for a moment
jim_thompson5910
  • jim_thompson5910
do you see how I got that domain?
ineedhelp10
  • ineedhelp10
yeah i saw how now
ineedhelp10
  • ineedhelp10
im just confused on the range now
jim_thompson5910
  • jim_thompson5910
what is the smallest value in the domain?
ineedhelp10
  • ineedhelp10
0
jim_thompson5910
  • jim_thompson5910
plug that into C(g) to get ??
ineedhelp10
  • ineedhelp10
0 again
jim_thompson5910
  • jim_thompson5910
so 0 is the smallest number in the range
jim_thompson5910
  • jim_thompson5910
what is the largest number in the domain?
ineedhelp10
  • ineedhelp10
its 12
jim_thompson5910
  • jim_thompson5910
if g = 12, then C(g) = ??
ineedhelp10
  • ineedhelp10
$30 ?
jim_thompson5910
  • jim_thompson5910
so the range spans from 0 to 30 (including both endpoints)
ineedhelp10
  • ineedhelp10
how will that look like when writing it down?
jim_thompson5910
  • jim_thompson5910
well you can either say \[\Large 0 \le y \le 30\] or \[\Large 0 \le C(g) \le 30\]
ineedhelp10
  • 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
  • jim_thompson5910
I'm glad to be of help

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