Final answer:
A mapping represents a function if each input maps to exactly one output. The vertical line test can be used to determine if a graph represents a function. A diagram can be modified to show that a mapping is not a function by indicating an input that maps to multiple outputs.
Step-by-step explanation:
In mathematics, particularly in the subject of algebra, a function is defined as a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. An easy way to determine whether a mapping represents a function is to apply the vertical line test. If any vertical line intersects the graph more than once, then the graph does not represent a function because that means an input (x-value) is associated with more than one output (y-value).
When attempting to justify whether a given mapping represents a function, we need to look for any indications that a single input maps to multiple outputs. This would invalidate the definition of a function. For instance, if we have an input x that maps to two different outputs y1 and y2, then we no longer have a function because a function requires that each input have one and only one output.
Modification of Diagram
To modify a diagram to show that a mapping is not a function, you would add another arrow from one of the inputs to a different output, thus showing that there is more than one output for a single input. However, without the specific details of the mapping in question or the options available for modification, it is not possible to provide a precise modification.