Question

Why is it impossible for a function to have fewer values in its domain than in its range?

Answer 1

It is impossible for functions to have fewer values in its domain than its range because from the definition of a function, 
             A function is an equation wherein for every independent variable there is one and only corresponding dependent variable. 

For example, for x = 1 the value of y = 0. It is also possible for x = 2 to have a value of y = 0. However, it is impossible for x = 1 to have a value of y = 0 and 1.