Question

How do you find the domain and range of `f(x)= x^2/(1-x^2)`?

Answer 1

The domain is `D_f(x)=RR-{-1,1}`
The range is `f(x) in (-oo,-1) uu [0,+oo)`

Explanation:

`f(x)=x^2/(1-x^2)=x^2/((1-x)(1+x))`

As we cannot divide by `O`, `x!=1` and `x!=-1`

The domain of `f(x)` is `D_f(x)=RR-{-1,1}`

To calculate the range, we need to calculate `f^-1(x)`

Let `y=x^2/(1-x^2)`

We interchange `y` and `x`

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

Now, we calculate `y` in terms of `x`

`x(1-y^2)=y^2`

`x-xy^2=y^2`

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

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

`y=sqrt(x/(x+1))`

The domain of `y` is the range of `f(x)`

What is underneath the `sqrt` sign is `>=0`

Therefore,

`x/(1+x)>=0`

We build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `-1` `color(white)(aaaaaaaa)` `0` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x+1` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f^-1(x)` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f^-1(x)>=0` when `x in (-oo,-1) uu [0,+oo)`