How to find the range of #x^2/(1-x^2)#?

Question

I thought that the answer is #R:(-oo,-1)uu(-1,+oo)#but the book says that the correct answer is #R:(-oo,-1)uu[0,+oo)#
Can anyone help please?

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Answer 1 · 2018-08-10T21:12:05

The range of #x^2/(1-x^2)# is #(-oo, -1) uu [0, oo)#

Let:

#y = x^2/(1-x^2)#

and solve for #x#...

Multiplying both sides by #1-x^2#, we get:

#y-yx^2 = x^2#

Adding #yx^2# to both sides, this becomes:

#y = (y+1)x^2#

Then dividing both sides by #(y+1)# we get:

#x^2 = y/(y+1)#

This has solutions if and only if:

#y/(y+1) >= 0#

That is, if either of the following:

So the range of #x^2/(1-x^2)# is #(-oo, -1) uu [0, oo)#

graph{x^2/(1-x^2) [-10, 10, -5, 5]}