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A function can map each element in the domain to only one element in the codomain or range. A relation is not so restricted.
A simple non-mathematical illustration:
relation: y = biological parent(x)
function: z = biological mother(x)
Leaving aside complications from surrogacy or other exceptional situations, each person has only one natural mother. Siblings may share the same natuarl mother but they are different elements of the domain.
However, for each person, there are two biological parents. The relationship or mapping is said to be one to many, and is therefore not a function.
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When can you say that the graph is function or mere relation?
If a vertical line intersects the graph at more than one point then it is not a function.
Is the relation a function?
Not every relation is a function. But every function is a relation. Function is just a part of relation.
How will you determine if a graph is a function or a mere relation?
You use the "vertical line test". If anywhere you can draw a vertical line that goes through two points of the graph, the relation is not a function; otherwise, it is a function. This is just another way of saying that in a function for every x value (input) there is AT MOST one y value (output).
What is the function of a mere?
To hold water. A mere is a lake.
Is mere relations a function?
No.
