Explanation:
a function is a map assigning values of a result or solution set to values of a set of initial or starting values.
we typically name these starting values x and the solution values f(x) or y (the results "created" out of x).
we often say
y = f(x)
and provide an expression or table to explain the logic of the mapping between x and f(x).
the only rule to be a function and not just a general relationship between these 2 sets :
every value in the set of starting values must have exactly one assigned solution value. not 0, not 2 (or more). one and only one.
but yes, different values of x can have the same assigned solution value. that is not a problem for a function. it just cannot be the other way around. there cannot be different solution values assigned to the same value of x.
so, a typical example of a function is a line.
e.g.
y = f(x) = 5x + 12
for every value of x you could think of (and everything else) there is exactly one result value calculated.
or the score board in a sports event :
athlete 1 : 12 points
athlete 2 : 24 points
athlete 3 : 20 points
athlete 4 : 22 points
...
the athletes are the "x". and their associated points the "f(x)" or "y".
no athlete is listed multiple times with different scores.
a typical example of not being a function is a circle. for almost every value of x inside the circle (that means for almost every point of the horizontal diameter of the circle) there are 2 values of y (that means points on the arc of the circle) : one above and one below the diameter.
or a strange score board like
athlete 1 : 12 points
athlete 2 : 24 points
athlete 3 : 20 points
athlete 2 : 22 points
athlete 4 : 22 points
...
athlete 2 is listed twice with 2 different scores.
so, this is therefore not a function.