Question

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

Answer 1

You first look for restriction in the domain. In this there are none, since the denominator will always be at least `1`

Explanation:

The minimum value this function can reach is when
`x=0->f(x)=(-1)/1=-1`
The maximum is when `x` gets really large, the function will be nearing `(2x^2)/x^2=2` without reaching it: `lim_(x->oo) f(x)=2`
So the range is `-1<=f(x)<2`
graph{(2x^2-1)/(x^2+1) [-10, 10, -5, 5]}