Question

I need to find the domain and range, x and y intercepts, horizontal and vertical asymptotes. Can someone help me through it please?

Answer 1

Hi
what is your function?

Answer 2

There should be a picture attached but it's:
X^2 + x - 2 divided by x^2 - 3x - 4

Answer 3

first thing, you should factor the num and denom of your function\[f(x)=\frac{x^2+x-2}{x^2-3x-4}\]have you tried to do so?

Answer 4

I did, the numerator factors into (x-1)(x+2) and the deniminator factors into (x-4)(x+1)

Answer 5

very good, so what will be the domain considering the denom?

Answer 6

The domain would be all reals except x=4 and x=-1?

Answer 7

quite right

Answer 8

let's work on range of the function

Answer 9

alright

Answer 10

Is the range all reals?

Answer 11

thats right, do you have a reasoning for that?

Answer 12

Well by the graph it extends infinitely up and down

Answer 13

and because quadratic functions' range will always be all reals

Answer 14

that's right, but what if we don't know about the graph

Answer 15

note that quadratics don't cover all reals

Answer 16

Oh okay, then I do not a reasoning

Answer 17

do not have a reasoning*

Answer 18

function is continuous on the interval (-1,4) and\[\lim_{x \to -1^{+}} f(x)=\infty \]\[\lim_{x \to 4^{-}} f(x)=-\infty \]therefore range is all of real numbers

Answer 19

does that make sense?

Answer 20

So when you plugged in the ordered pair, the output was infinite basically?

Answer 21

thats right

Answer 22

okay that makes sense

Answer 23

for example\[\lim_{x \to -1^{+}} f(x)=\frac{1 \times -2}{-5 \times 0^{+}}=\infty\]

Answer 24

ok what are x and y intercepts?

Answer 25

Alright so the y intercept = 1/2 because you plug in 0 for x to find y. The x intercept = -2,1 because they are the zeros of the numerator

Answer 26

very right, thanks

Answer 27

and what about asymptotes

Answer 28

The horizontal asymptotes is 1 because the degree of each function are the same and therefore, must divide the leading coeeficients. The vertical asymptotes are y= -1,4 because they are the zeros of the denominator.

Answer 29

right, you can put your words in mathematical phrases

Answer 30

now you can graph your function

Answer 31

Okay so for the asympotes I put dotted lines to represent them, and then I plugged in points according to get this graph Poor drawing but the basic jist of it :/