# Domain/Range of a function

## Main Question or Discussion Point

What function has a domain consisting of all real numbers and a range from [0,1]?

what kind of functions is it?

for example, if its an identity function, you can say "y=x for all x from 0 to 1"

also, if its a piecewise function, you can say "sin(x) for all x from 0 to pi, and sin(-x) for all x from pi to 2pi"

I`m just answering the question for fun, and sure to help u; but wait others for more reliable answers

There are many you can define.

f(x) = x - floor(x) is one.
g(x) = ceiling(x) - x is another.

(notice that f(x) + g(x) = 1, except perhaps in the case of x an integer, depending on the particular definitions of floor and ceiling you're using).

Other examples?

h(x) = |sin x|
i(x) = |cos x|
j(x) = (sin x)^2
k(x) = (cos x)^2

Not feeling like a periodic answer?

(l_n)(x) = exp(-x^2n), for any even integer n, defines a family of such functions.

(m_q,n)(x) = x^n / (q+x^n), for any even integer n and any positive real number q

Looking for something more exotic?

let n(x) be the function which equals 0.5 if x is rational, and 1.0 otherwise.

let o(x) be the function which gives the probability of two events both occurring if they are independent and have probabilities o(x/2) and o(2x).

(just for fun, could somebody find a closed-form solution for this last function, if there is one? does it make sense, or is it missing a necessary "base case"? one can tell that that o(0) = 1, but... can the rest be found uniquely?)

So, in response to the OP's question... pick your favorite.

Lol, nevermind about that last question.

It turns out the only such function is o(x) = 1...