Question

HELP PLEASE!!!!

When looking at a rational function, Charles and Bobby have two different thoughts. Charles says that the function is defined at x = −2, x = 3, and x = 5. Bobby says that the function is undefined at those x values. Describe a situation where Charles is correct, and describe a situation where Bobby is correct. Is it possible for a situation to exist that they are both correct? Justify your reasoning

Answer 1

@RadEn Can you help?

Answer 2

Hmm well coming up with a function that makes Charles correct seems fairly straight forward.

Can you think of a function that is defined at x=-2, 3 and 5? ( meaning: we are allowed to plug these values in for x).

Answer 3

5-2 = 3?

Answer 4

Hmm that doesn't contain any variables :c we wouldn't consider that a function.

How about something like:\[\Large\bf\sf y=x\]Are there any restrictions on this function? Any x values that we `can't` plug in?

Answer 5

Just in case I'm confusing you:\[\Large\bf\sf y=\sqrt x\]We can't plug negative numbers into this function, correct?
Those x values are not allowed.

So this is an example that would NOT work for Charles.
The previous one I mentioned would though.
Do you understand the difference?

Answer 6

thank you for the other explanation cause i was getting a bit confused, but yes i do see the difference now. so y=x would be the function for charles?

Answer 7

Yes that would work fine for Charles.
Bobby's function will take a little bit more work.

Answer 8

The key words here are RATIONAL FUNCTION, DEFINED AND UNDEFINED.
A rational function in its simplest form consists of one polynomial divided by another:
f(x)
y = ---- . Let's use that as the model for both Bobby's and Charles' rational function.
g(x)

Answer 9

@zepdrix you there? I still need help with Bobby's function

Answer 10

@mathmale i'm kinda confused.

Answer 11

If Bobby is right in saying that this rational function is undefined at {-2, 3, 5}, then this rational function has for its denominator (x+2)(x-3)(x-5).

Answer 12

ok, so what would the numerator be for that function?

Answer 13

f(x) Something
y = ---- is now equal to y = ------------------
g(x) (x+2)(x-3)(x+5)

Answer 14

ok i get it, are we supposed to find that something out?

Answer 15

Great question! My face suddenly turned red as I realized I'm uncertain about what to do next. What are YOUR thoughts?

Answer 16

I think maybe that the something could be anything, because to make the function true we just need the denominator to be (x+2)(x-3)(x+5)

Answer 17

wait, first you wrote x-5 and then x+5 which is it?

Answer 18

f(x) Something C(x+2)(x-3)(x+5)
y = ---- is now equal to y = ------------------ = ---------------------
g(x) (x+2)(x-3)(x+5) (x+2)(x-3)(x-5)

This is rather trivial in that it reduces to the constant, C.

Maria, glad you found my typo. Since the function is supposedly undefined at 5, the correct factor in the denom. is (x-5). Sorry about that.

Answer 19

This rational function, while "trivial," seems to fill the bill. As Bobby says, it's undefined at {-2,3,5}; if we reduce the function, we get y = C, which, as Charles says, is defined. Can anyone come up with a more sophisticated result for that "Something"?

Answer 20

while we find out something, is it possible for both of them to be correct? thats the last part of this question.

Answer 21

I think it's a matter of interpretation. The two kids are debating whether or not the same rational function (which was a given) is or is not defined at {-2,3,5}. If they don't reduce this function by cancelling like terms, then Bobby is right: the function is undefined at those x values. If they DO reduce it, then Charles is right in that the function is defined. This is a hypothetical situation. Hypothetically, if those conditions are met, either boy could be "right." What do YOU think?