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

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]}