To solve the problem and determine which relations represent functions, we apply the definition of a function: A relation is a function if every input (x-value) is associated with exactly one output (y-value). Let us evaluate each relation provided:

1. First Relation (Mapping Diagram):

   • $x = -7 \to y = 0$
   • $x = -1 \to y = 0$
   • $x = 0 \to y = 6$
   • $x = 5 \to y = 6$
   • Each $x$-value maps to exactly one $y$-value. This is a function.

2. Second Relation (Graph):

   • A vertical line test shows that this graph fails the test because one vertical line intersects the graph at two points. This is not a function.

3. Third Relation (Graph):

   • A vertical line test shows that any vertical line intersects the graph at most one point. This is a function.

4. Fourth Relation (Mapping Diagram):

   • $x = 2 \to y = 4$
   • $x = 7 \to y = -1, 3$ (one $x$-value maps to multiple $y$-values)
   • $x = 11 \to y = 8$
   • Since $x = 7$ maps to two different $y$-values, this is not a function.

Final Answer:

• The relations that represent functions are:
  • First Relation (Mapping Diagram)
  • Third Relation (Graph)

Would you like me to clarify further or explore related concepts? Here are some follow-up questions for deeper understanding:

1. What is the vertical line test, and how does it work to identify functions?
2. Can a mapping diagram include repeated outputs (y-values) and still represent a function?
3. Why is it important for each input to have a unique output in functions?
4. How would this problem differ if we were testing for injective or surjective functions?
5. Can a relation be a function if no inputs are repeated but the same output occurs multiple times?

Tip: When checking graphs for functions, always use the vertical line test as a quick visual check.