1. ## domain and range

am i right?
$y=-\sqrt{x^2+2}$

domain: $x|x\ge-2$
range;?

how can i graph this?
thanks.

2. Originally Posted by princess_21
am i right?
$y=-\sqrt{x^2+2}$

domain: $x|x\ge-2$
range;?

how can i graph this?
thanks.
---------------------------
domain: $x|x\ge-2$

Not exactly

This will give real values if

x^2 + 2 > 0

Since Square of every real number is >0
And +2 will not make it negative

Hence its true for all real x
------------------------------------------------
Range:

Thus
x^2 = y^2 - 2

$x= \pm \sqrt{y^2 -2}$

For rage you need to find all y that satisfy above values

thus

$y^2 \ge 2$

So all y EXCEPT ( -2,2)
But y should be negative because of the question's negative sign
so range is
[2,infinity)

------------------------------------

Graph:
Note that y will always be negative

y^2 = x^2 + 2

y^2 -x^2 = 2

This is a 90degree rotated hyperbola (Vertical hyperbola)

the upper part of vertical hyperbola should not be drawn

The graph below is for

Again Don't draw the upper leaf of this type of hyperbola

---------------------------
domain: $x|x\ge-2$

Not exactly

This will give real values if

x^2 + 2 > 0

Since Square of every real number is >0
And +2 will not make it negative

Hence its true for all real x
------------------------------------------------
ok I understand this part. thanks

Range:

$(y)^{2}=(-\sqrt{x^2+2})^{2}$

$y^2=-x^{2}-2$

$x^2=-y^{2}-2$

how did you get x^2 = y^2 - 2

Thus
x^2 = y^2 - 2

$x= \pm \sqrt{y^2 -2}$

For range you need to find all y that satisfy above values

thus

$y^2 \ge 2$

So all y EXCEPT ( -2,2)
But y should be negative because of the question's negative sign
so range is
[2,infinity)
I didn't understand the range can you explain further? thanks so much

4. Originally Posted by princess_21
ok I understand this part. thanks

I didn't understand the range can you explain further? thanks so much
First of all there was a TYPO at the last step, corrected here (written y as negative and gave you +ve range)

$
\implies y^2 = (-1)^2 \times (\sqrt{x^2 +2})^2$

$y^2 = 1 \times (x^2 +2)$

Now

y^2 - 2 = x^2

And

Now this represents x in terms of y

Domain of x is R

what are the values that y can take to give all x in R

This is done by finding all the values in this function

in other words we find the domain(meaning all values that y can take) of this function

Hence I followed the steps below

-----------------------------------------------------

thus

So all y EXCEPT ( -2,2)
-----------------------------------------------------
But y should be negative because of the question's negative sign
this happens because

has LHS negative (-1 x square root= negative number)

so range is

(-infinity,-2]