Question

Bonnie buys a used car for $2500. She calculates her average monthly expenses for gas, insurance, and maintenance at $150 per month. The function f(x) = 150x + 2500 represents her total cost of owning the car for x number of months.

Answer 1

What is the domain?

Answer 2

Well, can the domain be negative?

Answer 3

I should say - can x be negative?

Answer 4

No, can't be negative.

Answer 5

OK, so we know x must be non-negative.

Can x=0?

That one is kind of tricky... could be a matter of interpretation, lol. Since she BOUGHT the car already, she owns it for at least SOME PART of a month, even if she drives down the street and sells it. So I think I'd say x must be strictly positive.... agree?

Answer 6

Yes, I agree.

Answer 7

OK, so we have 0<x. The only question not is the upper bound on the domain. Is there one? is there a maximum number of months that she can own the car?

Answer 8

No so it would be greater or than or equal to.

Answer 9

No, we've already settled the sign at the lower end, e.g., can't have x=0, so 0<x.

The question now is an upper bound. E.g., if the car will self-destruct if she owns it for 3 years or more, then:

0<x<36

So what do you think about an upper bound?

Answer 10

Would it be 2500?

Answer 11

The domain would be x(greater than or equal to) 2500, correct

Answer 12

why would it be 2500??

Answer 13

No...... x>0. We already know what x is GREATER than. Remember what x is: the number of months she owns the car. It has nothing to do with the money.

Answer 14

I thought due to the cost? That's the max or total what was paid.

Answer 15

Oh ok, so the domain is x is ( equal to or greater than) 0

Answer 16

Maybe we should back up. Do you know what "domain" means?

Answer 17

I do for the most part, just can't explain it. I'm sorry

Answer 18

I apologize, as the domain will be a 'y' answer, I believe.

Answer 19

From what I know the domain should be y is (equal to or greater than ) 0. I believe this is correct.

Answer 20

I'm not sure what you mean by that, but let's start with domain.

A function has "inputs", the x-values, and "outputs", he y-values. The domain is the collection of all the possible (allowable) x's.

So for example, in an equation like y=x-1, the domain is all real numbers.

But in \[y=\frac{ 5 }{ x }\] you can't have x=0, so you have to leave it out of the domain. The domain there is all real numbers EXCEPT x=0.

So in your problem, the equation mathematically is just a linear equation, which is well-defined for all values of x. That is, if it wasn't a word problem and I just said "what's the domain of the function y=150x+2500", the answer would be "all real numbers".

But I presume that here, they want you think about the CONTEXT of the function to determine the domain. x is the time she owns the car, so that can't be negative, and as we discussed above, can't =0. Thus 0<x.

Now you could say "well, she can't live forever and the car can't last forever, so there is an upper bound on x also". And while that's technically true.... what do you pick as that upper bound? 10 years? 20 years? 50 years?? There's really no legitimate way to pick an upper bound here, so I would say that the domain is:\[0<x<\infty \]or depending on the notation that you are asked for, you could write as an interval:\[\left( 0,\infty \right)\]

Answer 21

Ok so I was correct x is ( equal to or greater than) 0? I thought this was correct, but just wanted to be sure.

Answer 22

Well, as I said above, I think whether x=0 is allowable is open to interpretation, but I would say NO. x is the number of months she owns the car. Is it possible that she won't own the car at all? She already bought it - so even if she sells it the same day, it won't be x=0. So I think x is strictly greater than 0

Answer 23

It, the question only gives me 4 possible answers. x is ( greater than or equal to ) 0, x is (greater than or equal to) 2500, or both the same answers with y?

Answer 24

OK, well, if it's multiple choice then then it would definitely be the 1st one. Domains don't have anything to do with y (that's the range). And the domain here certainly has nothing to do with the 2500. :) So in their interpretation, you can have x=0 and while I might disagree with that on a philosophical level, it is clearly the best answer (x> or =0)

Answer 25

Thank you, Debbie for your help!! :)

Answer 26

You're very welcome @lorilynn :)