Question

f(x)=(3x^2 + x - 5)/(x^2 + 1)
Find the domain of the function, decide if the function is continuous, and identify any horizontal and vertical asymptotes.

Answer 1

Help please!!!

Answer 2

@greenglasses sorry to ask for your help again, my teacher never taught us how to do these kinds of questions.

Answer 3

I don't believe you can simplify that.

Answer 4

Because you can't simplify it, there are no vertical asymptotes, and the domain must be xER

Answer 5

Does xER mean all real numbers x?

Answer 6

Do you know how to divide equations?

Answer 7

Yeah, it's the closest that I could find on a keyboard.

Answer 8

Yeah, you mean by long division?

Answer 9

Yes. Can you try to divide this equation?

Answer 10

I got 3 + (x-8)/(x^2+1)

Answer 11

Looks good to me. Alright, I'm gonna give you a crash course on limits.

Answer 12

If there's a horizontal asymptote, as x becomes a bigger and bigger number, it'll approach a certain number.

Answer 13

Let's say that x is a super big number, like 1000000. If you plug that into the equation, -8 and +1 become negligible, right?

Answer 14

So all you have to worry about is \[\frac{ x }{ x^2}\] , really. So if x were a really big number, what would the above equation be equal to?

Answer 15

I'm not sure... The answers in the back of the book say 3... But I don't understand.

Answer 16

Yes, it is three

Answer 17

See, \[\frac{ 1 }{ 100000000 }\]
is really close to zero, right? it's 0.0000...1

Answer 18

Since we're adding it to three (because 3 + (x-8)/(x^2+1)), then that means that when x is really big, it approaches 3.

Answer 19

That means that the horizontal asymptote is at x=3

Answer 20

*y=3

Answer 21

So the number you add it to when you've finished long division is the horizontal asymptote?

Answer 22

Most always yes.

Answer 23

What are some cases when it would not?

Answer 24

And how can I tell if it's continuous or not?

Answer 25

I can't think of any situations off the top of my head, but it's always best to assume that it's not always the case, and I can recall running into a couple situations where it wasn't.

Answer 26

A function if continuous basically if it doesn't have a hole or a vertical asymptote.

Answer 27

There's a more official definition but it's a lot more confusing and you don't really need to know it.

Answer 28

Since this one doesn't have a hole or vertical asymptote it is continuous?

Answer 29

Yes.

Answer 30

Thanks again!

Answer 31

Np.