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

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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?

I thought that the answer is

Can anyone help please?

The range of

Let:

This has solutions if and only if:

That is, if either of the following:

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

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To find the range of the function ( f(x) = \frac{x^2}{1 - x^2} ), we need to consider the restrictions on the domain that ensure the function is well-defined.

The denominator ( 1 - x^2 ) cannot be equal to zero because division by zero is undefined. Therefore, we must find the values of ( x ) that make ( 1 - x^2 ) equal to zero.

Solving ( 1 - x^2 = 0 ) gives us ( x^2 = 1 ).

This equation has two solutions: ( x = 1 ) and ( x = -1 ).

Thus, the function is undefined when ( x = 1 ) and ( x = -1 ).

Next, let's consider the behavior of the function as ( x ) approaches these values.

As ( x ) approaches ( 1 ) or ( -1 ) from the left, ( 1 - x^2 ) becomes positive, and as ( x ) approaches ( 1 ) or ( -1 ) from the right, ( 1 - x^2 ) becomes negative. This means that ( f(x) ) approaches positive infinity as ( x ) approaches ( -1 ) from the left, and negative infinity as ( x ) approaches ( -1 ) from the right. Similarly, ( f(x) ) approaches positive infinity as ( x ) approaches ( 1 ) from the right, and negative infinity as ( x ) approaches ( 1 ) from the left.

Therefore, the range of the function ( f(x) = \frac{x^2}{1 - x^2} ) is all real numbers except ( -\infty ) to ( -1 ), ( 1 ), and ( 1 ) to ( \infty ). In interval notation, the range is ( (-\infty, -1) \cup (-\infty, 1) \cup (1, \infty) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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