Question

Range of 1/[root(x^2-4)]

I solved in following way.. kindly check its right or wrong
First, I converted x in terms of y as follows

x=root[(1+4y^2)/y^2]
f^-1(y)=root[(1+4y^2)/y^2]
f^-1(x)=root[(1+4x^2)/x^2 ]
Where (1+4x^2)/x^2>=0
This inequality is true for any real values of x. So, its range should not be (- infinity ,+ infinity)

Answer 1

You might think that, but f(x) does not have a single inverse, since it has two branches. That is, it fails the horizontal-line test.
In either case, x=0 must be excluded, since you are dividing by x^2.
In addition, you must pick one of the branches, so the range is either
(-∞,0) or (0,∞)