Question

Which one of the functions always maps the values between 0 and 1

Answer 1

There are many function which will map the values from 0 to 1
Let us take an example for that

Let f(x)=1/x
Where the domain of x is from 1 to ∞
So the output of the function will also be mapping from 0 to 1

The following function created is also one-one, which means for any unique value input the output will also be unique
No , two input values will have the same output
Now,
Let us take another example of the function whose value will map from 0 to 1
Example:

f(x)=mod(sin(x))
So the input will vary from -∞ to ∞
But the output will always be between 0 to 1

Now the above function is not one-one since the input value of 0 and pi will have same output value
Similarly is,
f(x)=mod(cos(x))

These are some of the functions whose output value is from 0 to 1 both inclusive