Question

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?

Answer 1

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]}