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

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?

1 Answer
Aug 10, 2018

Answer:

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

Explanation:

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:

  • #y >= 0" "# and #" "y + 1 > 0#. That is #y >= 0#

  • #y <= 0" "# and #" "y + 1 < 0#. That is #y < -1#

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]}