# Domain/Range of a function

#### cahiersujet

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

#### wajed

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

#### csprof2000

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.

#### csprof2000

Lol, nevermind about that last question.

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

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