Find a rational function f: R -R with range f(R) = [0,1]. (Thus f(x) = p(x)/q(x) for all x for suitable polynomials P and Q where Q has no real roots.

Question

Find a rational function f: R ->R with range f(R) = [0,1]. (Thus f(x) = p(x)/q(x) for all x for suitable polynomials P and Q where Q has no real roots.

zarkon · Answer

it doesn't look like this function attains the value 1

zarkon · Answer

$$f(x)=\frac{2x^2}{x^4+1}$$

anonymous · Answer

Zarkon's answer is correct.

anonymous · Answer

thanks, how did you obtain this though? have to explain it quite a bit.

zarkon · Answer

experience

anonymous · Answer

can you explain further?

anonymous · Answer

you want q to have no real roots so some obvious choices are 
$$x^2+1,x^4+1, x^2+2,...$$

anonymous · Answer

want range to be [0,1] so you can insure it is zero at zero by making the numerator 
$$x, x^2, 2x^2, ...$$