I need help with domain

Question

Assume a car uses gas at a constant rate. After driving  25 miles since a full tank of gas was purchased, there was 14 gallons of gas left; after driving  65 miles since a full tank of gas was purchased, there was  10.8 gallons of gas left.

Use a function to model the amount of gas in the tank (in gallons). Let the independent variable be the number of miles driven since a full tank of gas was purchased. Find this function’s domain and range in this context.

My teacher is no help I genuinely do not understand

Answer 1

distance=(# of gallons used)*(miles/gallon)

let g=# of gallons in the tank when full

25=(g-14)(m/g) m=# of miles g=# of gallons

65=(g-10.8)(m/g)

25g=(g-14)m

25g/(g-14)=m

65g=(g-10.8)m

65g/(g-10.8)=m

notice there are two equations equal to "m"

set them equal to each other

25g/(g-14)=65g/(g-10.8)

cross multiply

25g(g-10.8)=65g(g-14)

25g^2-270g=65g^2-910g

65g^2-25g^2-910g+270g=0

40g^2-640g=0

divide by 40

g^2-16g=0

factor

g(g-16)=0

g-16=0

g=16

the tank holds 16 gallons of gas

from the information given...

you travel 25 miles on 2 gallons of gas and travel 65 miles on 5.2 gallons of gas

25/2=12.5 miles per gallon and 65/5.2=12.5 miles per gallon also

miles driven(m)=gallons used(16-g) * 12.5mpg

so, m=(16-g)*12.5

you want m to be the independent variable

m/12.5=16-g

g=16-(m/12.5) (answer)

plug in 25 for m and plug in 65 for m to see that this is correct-that you get the amount of gas left in the tank

the domain of m is 0≤m≤200

the range is 0≤g≤16