Question

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

Answer 1

x+1/absx+2

Answer 2

could you explain it please? thankyou!

Answer 3

you could try something like
\[f(x)=\frac{x}{x^2+1}\]

Answer 4

i mean to say something "like" it. that one doesn't work because the range of
\[f(x)=\frac{x}{x^2+1}\] is \([-\frac{1}{2},\frac{1}{2}]\) you will have to adjust it

Answer 5

the first response isn't a rational function. They have to be polynomials.

Answer 6

only problem with @mahmit answer is \(|x+2|\) is not a polynomial

Answer 7

oh what @scarydoor said

Answer 8

@satellite73 's hint is on the money... easy to convert that to the right function.

Answer 9

@scarydoor how can i convert it to the right function? i dont understand

Answer 10

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

can anyone confirm this answer? i think its right...

Answer 11

@satellite73

Answer 12

Satellite's function is almost right, in that the range is [-1/2, 1/2]. But you want it [-1,1]. So you want to stretch it out to that. If you multiply the function by 2, then if you think about it a bit, you'll see that the range will be [-1,1].

Answer 13

ahh thankyou! yes it makes sense