Question

PLEASE HELP!!!
What is the Domain and Range for the function y=-sec x

Answer 1

hint:

sec(x) = 1/cos(x)

Answer 2

look for possible places where you'll divide by zero (so you know which values to exclude from the domain)

Answer 3

what do you mean divide by 0?

Answer 4

dividing by zero is undefined, so something like 1/0 is not allowed

Answer 5

if you can find the values of x that make cos(x) equal to zero, then you'll be able to exclude those values from the domain

Answer 6

so what values of x make cos(x) = 0 true?

Answer 7

90 degrees?

Answer 8

cos is x and at 90 degrees the coordinate is (0,1)

Answer 9

x = 90 is just one of infinitely many x values that make cos(x) = 0 true

Answer 10

x = 270 is another x value that makes cos(x) = 0 true because on the unit circle, we're at (0,-1)

Answer 11

so how do i write that in the Domain?

Answer 12

Do you see the pattern that generates the x values that make cos(x) = 0 true

Answer 13

is it every 180 degrees?

Answer 14

that's part of it

Answer 15

what else?

Answer 16

all real numbers?

Answer 17

no

Answer 18

im not so sure...

Answer 19

you were on the right track when you said "every 180 degrees"

Answer 20

what's your starting value?

Answer 21

0 right?

Answer 22

no x = 0 makes cos(x) = 1

Answer 23

we want cos(x) = 0

Answer 24

oh so 90 degrees

Answer 25

you start at 90 and you add on multiples of 180

Answer 26

that's saying 90 + 180n where n is an integer

Answer 27

ok so x=90+180 how would you make it multiples

Answer 28

try it out: plug in n = 0 and you should get 90

plug in n = 1, you should get 270
etc etc

Answer 29

those values make cos(x) = 0 true

Answer 30

is it the x=90+180 pi k thing?

Answer 31

k is often used instead of n, so yes

Answer 32

wait what about it being -sec(x)

Answer 33

so effectively, we start with all real numbers as the domain BUT we kick out numbers like 90, 270, etc etc

Answer 34

so we have a set of all real numbers, but x cannot equal 90+180k where k is an integer

Answer 35

the pi shouldn't be there (since we're in degree mode)

Answer 36

ok but if we switch it to radian mode it would be pi/2+180n what about it being -sec(x)?

Answer 37

no if we switch over to radians, then you have to turn 180 into pi

Answer 38

since 180 degrees = pi radians

Answer 39

the domain of sec(x) is the same as -sec(x)

Answer 40

oh so x=90+pik?

Answer 41

you're mixing up the two modes

Answer 42

so the original problem is asking for -sec(x) but then -sec(x)=1/cos(x) right so theyre the same but inverses right?

Answer 43

-sec(x) = -1/cos(x)

Answer 44

we're still dividing by cos(x), so that's why the domain is the same as sec(x) = 1/cos(x)

Answer 45

they aren't inverses of each other though

Answer 46

oooh i see so the final answer is just x=90+180k

Answer 47

for the Domain

Answer 48

if you're in degree mode, yes and x cannot equal 90+180k

k is an integer

Answer 49

x can be any other number you want, but x cannot equal 90+180k

Answer 50

ok so thats not hard! how do you find the Range?

Answer 51

what is the range of cos(x)

Answer 52

im not sure how to find that

Answer 53

well plug in values of x into cos(x) and tell me what you get

values like x = 0, x = 45, x = 90, etc

Answer 54

cos(0)=1 cos(45)=square root 2/2 cos(90)=0

Answer 55

so far, what's the smallest and largest output?

Answer 56

0 smallest and 1 biggest right?

Answer 57

it turns out that 1 is the largest, but 0 isn't the smallest

this is because cos(x) can get negative for some x values

Answer 58

ok so range is something like 0>x<1

Answer 59

more like -1 <= y <= 1

since -1 is the smallest y = cos(x) can go

Answer 60

how do you know its exactly -1 is the smallest it can go

Answer 61

because on the unit circle, x = -1 is the furthest to the left you can go

Answer 62

oooh i see thank you so much i need some practice but i think i get it!!!

Answer 63

so if cos(x) gets closer to 0, then what happens to -1/cos(x)