Question

y= the square root of x^2-4....... how do i dertermine the domain and range

Answer 1

\[y=\sqrt{x^2-4}\]

Answer 2

for domain...under radical must be greater than or equal to zero or we can say it cant be negative thats the restriction we must consider for the case of radicals so\[x^2-4\ge0\]

Answer 3

ok so 2 would be a possiblity for x

Answer 4

do u know how to find all possible values for x from\[x^2-4\ge0\]?

Answer 5

yeah but theres so many options even negative options

Answer 6

@sarah70 exactly! you need to see for what values of x, x^2-4 goes negative

Answer 7

and remove them from domain

Answer 8

test this : x is between 0 and 2

for ex, x = 1 :
x^2-4
1^2-4
1-4
-3

so this 0<x < 2 should be removed from DOMAIN

Answer 9

how do you know if x is between 0 and 2?

Answer 10

here, how i did :

x^2-4

since 4 = 2^2, if i put x=2 above, it gives me 0

right ?

Answer 11

yes

Answer 12

you can see that anything more than 2 for x, makes x^2-4 a positive

Answer 13

does that make sense

Answer 14

yes

Answer 15

so, [2, infiniti) is part of our domain, okay

Answer 16

we searched 0 to +infinity
and we found those as domain.

we need to still search 0 to -infiniti, okay ?

Answer 17

isnt that also infinite?

Answer 18

you mean to say :

(-infiniti, -2]

?

Answer 19

huh? no i meant if you plug in all the negative values for x wouldnt the possiblities also be infinite?

Answer 20

oh ok thats right !! almost all -ve values of x, makes x^2-4 a positive, except few

Answer 21

test : x = -1

Answer 22

by now you should be able to see that :

domain of \(\sqrt{x^2-4} \ is\ x \le -2, x \ge 2 \)

Answer 23

anything between, -2 and 2 makes the thing inside radical negative !

those values are not fit for domain

Answer 24

but -1^2=1 and 1^2= 1 and theyre both greater than 0 so zero is the only # that doesnt apply so the domain is all real numbers except zero

Answer 25

but we need to see the *whole thing* inside radical :

x^2-4

Answer 26

put x = -1, above. what you get ?

Answer 27

oh NOW i understand

Answer 28

glad to hear :D

Answer 29

but isnt there a certain way to write it?

Answer 30

yeah you can express them in "interval notation"

Answer 31

domain : (\(-\infty \), -2] U [2, \(\infty\))

Answer 32

would i go about solving for the range the same way?

Answer 33

nopes

Answer 34

domain is about finding what all *input values* a function takes

Answer 35

range is about what all *output values* a function *gives out*

Answer 36

right ?

Answer 37

yup

Answer 38

so, just plugin the boundary values of domain, and see what the function gives out

Answer 39

we got the fomain as ,

\(x \ge 2\) :
\(x \le -2\) :

Answer 40

domain*

Answer 41

put x = 2, 3, 4....
you can see that \(\sqrt{x^2-4}\) gives out, 0 and higher values

Answer 42

put x = -2, -3, -4...
you can see that \(\sqrt{x^2-4}\) gives out, 0 and hight values AGAIN

Answer 43

higher*

Answer 44

do the range is \([0, \infty)\)

Answer 45

*so

Answer 46

ok thanks for your time that was very detailed thank you :)

Answer 47

yw !! i did these very long time ago... i enjoyed going thru these again. thanks you too :)