Question

HELP!! PLEASE!

f(x) = quantity x squared plus x minus two divided by quantity x squared minus three x minus four

Graph

A coordinate axis scaled by one.

Domain and Range (1 point): _____________________

x and y Intercept(s) (1 point): _____________________

Horizontal Asymptote(s) (1 point): ___________________

Vertical Asymptote(s) (1 point): ____________________

Answer 1

\(\Large\color{black}{ f(x)=\frac{x^2+x-2}{x^2-3x+4} }\) . right?

Answer 2

first find any domain restrictions (when x is equal to zero) and that will be our vertical asymptote(s).

Answer 3

yes

Answer 4

\(\Large\color{black}{ x^2-3x+4\ne0 }\) .

Answer 5

Im still I little confused, give me a minute.

Answer 6

Ok, so would our Vertical asymptote: 0?

Answer 7

im lost

Answer 8

can you solve for x, when x^2-3x+4=0?

Answer 9

oh ok

Answer 10

x^2-3x+4=0 (x-3) (x+4)

Answer 11

just got off a concussion sorry.

Answer 12

you can see that x^2-3x +4 can't be zero, because x^2-3x, for any smallest value of x, would have a lesser absolute value than 4.

Answer 13

So you don't really have an asymptote of a real number. I mean a vertical asymptote.

Answer 14

Okay, knowing that you don't have a vert. asymptote...

Answer 15

yea, so its 0 right

Answer 16

close lol. I would say, there is ZERo vertical asymptotes. Look over what I said, if you don't know why is that...

Answer 17

Tell me when you are good to proceed.

Answer 18

I just said, 0 vertical asymptotes, but ok proceed.

Answer 19

Do you know why there is no vertical asymptote? are you sure you want to go on?

Answer 20

yes, please go on.

Answer 21

Okay, so the domain, since there is no vertical asymptote - which removes any possible domain restriction, is all real numbers (you know what that means in an interval notation, right?)

The range however, we have to solve for.
\(\large\color{black}{ y=\frac{\LARGE x^2+x-2}{\LARGE x^2-3x+4} }\)

\(\large\color{brown}{ (x^2-3x+4)y= x^2+x-2 }\)

Answer 22

do you see just the math that I did so far?

Answer 23

I multiplied both sides times `x^2-3x+4` right?

Answer 24

oh ok.

Answer 25

I lost my latex work...

Answer 26

https://www.desmos.com/calculator/lq6z91wssn
TAKE A LOOK AT THIS GRAPH IF YOU WANT...

Answer 27

ok

Answer 28

ther eis your range, it goes from 0.5 to 2.094

Answer 29

No x intercepts, because the range never reaches zero.

Answer 30

oh ok

Answer 31

actually there is a bit lower point where range is 0.478 (a but lower than 0.5)
but the y-intercept is (0, 0.5)

Answer 32

there is no horizontal asymptote either.

Answer 33

would the horizontal asymptote be 0 or is there none at all?

Answer 34

oh ok

Answer 35

yes, none at all.

Answer 36

So what is the domain again