Question

how to find the domain and range of the equation in the attacked file below

Answer 1

For the vertical asymptotes, factor out x^5 from the numerator and the denominator and then take the limit as x-> +/- infinity

Answer 2

ya i got the asymptotes

Answer 3

For the rest of the questions you first need to factor the denominator and the numerator.

Answer 4

vert= x=3, x=-1, x=2
horizon= y=3

Answer 5

how do you factor it?

Answer 6

How did you find the vertical asymptotes before factoring the denominator?

Answer 7

i found the zeros

Answer 8

How?

Answer 9

oh i did factor it your right

Answer 10

(x+1)(x-3)(x-2)(x-2i)(x+2i)

Answer 11

now what?

Answer 12

You are still not telling me how you factored it.

Answer 13

i used synthetic division

Answer 14

that is the denominator only though

Answer 15

Did you use the rational roots theorem to find possible zeros?

Answer 16

find possible zeros

Answer 17

ya

Answer 18

You need to use the same method to factor the numerator. There may be cancellations and so you cannot rush to finding the vertical asymptotes just based on the denominator alone (although in many cases it will work)

Answer 19

how do you find the domain and range though?

Answer 20

Here the domain is all x values that will NOT make the denominator zero. Assuming you factored the denominator correctly, the domain is all real values of x except \(x \ne -1\) and \(x \ne 2\) and \(x \ne 3\).

Answer 21

before finding the range, factor the numerator and see if something cancels and simplifies the function.

Answer 22

(-(infinity),-1)(-1,2)(2,3)(3,(infinity))
is that the domain in interval form

Answer 23

\[
x = (-\infty, -1) \cup (-1, 2) \cup (2,3) \cup (3, \infty)
\]

Answer 24

of and where the verticle asymptotes cross the x int is that a x intercept point?

Answer 25

To find the y-intercept, set x = 0 and find y (or f(0)).
To find the x-intercept, set y = 0 and solve for x. (once again factoring numerator will help here).

Answer 26

for y int set the denominator or numerator to 0?

Answer 27

y intercept: set x = 0 in the whole function and then find y. In other words, find f(0).

Answer 28

and same for x int?

Answer 29

No. For the x-intercept, set y or f(x) = 0 and solve for x. This is usually when the numerator is zero. But you have to make sure there are not any factors in the numerator that cancels with the denominator.

Answer 30

ok now what is the range in interval notation

Answer 31

brb. phone call.

Answer 32

k

Answer 33

i have (x-1)(x+2)(x+3)(x-i)(x+1) as the numerator

Answer 34

After you factor the numerator we can look at the range.

Answer 35

the last factor is x+i

Answer 36

There should be a constant 3 too in the numerator.

Answer 37

im so confused

Answer 38

which part?

Answer 39

everything

Answer 40

You seem to have done well so far in finding all the factors.

Answer 41

im gonna attach a pic of my factors hold on

Answer 42

if you multiply all the x's the highest power you get is x^5 with a coefficient of 1. But in the numerator it is 3x^5. There should be a factor 3 outside.

Answer 43

Also, there is not need to factor that leads to imaginary factors. you can stop a step before that.

Answer 44

ok so in the denominator i just need to put a 3 on the outside

Answer 45

and thats it?

Answer 46

numerator

Answer 47

so now how do i find my range?

Answer 48

\[
\frac{3x^5 +12x^4 + 6x^3 - 6x^2 + 3x - 18) } { (x^5 - 4x^4 + 5x^3 - 10x^2 + 4x + 24) } =
\frac{3(x-1)(x+2)(x+3)(x^2+1)}{(x+1)(x-2)(x-3)(x^2+4)}
\]

Answer 49

and also my x and y inters

Answer 50

Once we factor the numerator and denominator we can answer all the questions. This should have been the first step: to factor BOTH denominator and numerator. Now we can answer each of the questions properly after noting there is NO common factor that cancels out.
a) vertical asymptotes. The ones that make the denominator zero: x = -1, x = 2 and x = 3
b) Horizontal asymptote: Both the numerator and denominator are 5th degree polynomials and so the horizontal asymptotes will be y = the ratio of the coefficients of the highest power: y = 3/1 = 3

Answer 51

c) Domain excludes point that make the denominator zero. So x is all real values except -1, 2 and 3.
d) Range: When x approaches each point of discontinuity (that is, x = -1, 2, 3), and if you take the limit, you will notice sometimes the limit is -infinity and sometimes it is +infinity. So the range is ALL real values. But the function is NOT defined when x = -1, 2 3.

Answer 52

so is the answer to d (-(infinity),(infinity))

Answer 53

yes.

Answer 54

e) x-intercepts means where does the graph of f(x) cut the x-axis? We know that on the x-axis, the y-value is zero. So set y = 0 and solve for x. y = 0 when the numerator is zero and that happens when x= -3, x = -2 and x = 1 and those are the x-intercepts.

Answer 55

f) e) y-intercept means where does the graph of f(x) cut the y-axis? We know that on the y-axis, the x-value is zero. So set x = 0 and solve for y. For this look at the function before factoring. put x = 0. y = -18/24 = -3/4

Answer 56

wait so the y intercept is (0,-3/4)

Answer 57

you can list that in the coordinate form as (0, -3/4) or you can say the y-intercept is -3/4 depending on what form is acceptable to your teacher/book.
But be consistent. If y-intercept is expressed as (0, -3/4) then you should probably express x-intercepts as (-3,0), (-2,0) and (1,0)
whichever format is acceptable to them.

Answer 58

ok that is what i did so lastly what are all the possible zeros

Answer 59

Oops. the denominator should be factors of 3.

Answer 60

not 24

Answer 61

g) Possible zeros. Use rational roots theorem to simply LIST all possible zeros. I am assuming they want the possible zeros only for the numerator (or maybe they want for both). But for the numerator the possible zeros are:\[
\pm \frac{\text{Factors of 18}}{\text{Factors of 3}} = \pm\frac{1,2,3,6,9}{1,3} \\
\]List all possibilities: -1, 1, -1/3, +1/3, -2, +2, -2/3, +2/3, -3, +3, -6,+6, -9, +9

Answer 62

am i done besides graphing?

Answer 63

if they want all the possible zeros what should i do?

Answer 64

yeah, i left out the factor 18 in the numerator.

Answer 65

so is it wrong?

Answer 66

It is correct.

Answer 67

assuming they want it only for the numerator.

Answer 68

http://assets.openstudy.com/updates/attachments/5421d64de4b022753f9a9c07-campbell_st-1411507525074-captureasymptotes24th.png

Answer 69

is that what the graph should look like

Answer 70

How did you get that graph? Did they give it to you?

Answer 71

no someone else on this website sent it to me

Answer 72

yes, that is the correct graph. It confirms almost all the answers we got from doing algebra.

Answer 73

thank you so much for all your help i really appreciate it! you have some awesome karma coming your way!

Answer 74

You are very welcome. Thank you.

Answer 75

For domain don't forget the Union symbol \(\cup\) in between subdomains in your answer sheet: \(\text{Domain = } (-\infty, -1) \cup (-1, 2) \cup (2,3) \cup (3, \infty)\)