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domain:

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

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Range:

Thus

x^2 = y^2 - 2

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

thus

So all y EXCEPT ( -2,2)

But y should be negative because of the question's negative sign

so range is

[2,infinity)

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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