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.

Answer 1

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

Answer 2

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

Answer 3

Zarkon's answer is correct.

Answer 4

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

Answer 5

experience

Answer 6

can you explain further?

Answer 7

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

Answer 8

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