To determine which table shows y as a function of x, we need to recall the definition of a function: for each input (value of $x$), there must be only one output (value of $y$). In other words, a function assigns exactly one output to each input.

Let’s analyze each table:

• a.

  • For $x = 4$, we have multiple $y$-values: $y = -3, 8, 10, 14$.
  • Since the same input $x = 4$ is associated with multiple different outputs, this does not represent a function.

• b.

  • Each $x$-value (2, 4, 6, 8) is associated with only one corresponding $y$-value (4, 8, 8, 16).
  • This represents a valid function since every input has exactly one output.

• c.

  • $x = -3$ appears twice with different $y$-values (4 and 8), which violates the rule of a function.
  • This does not represent a function.

• d.

  • Each $x$-value (0, 1, 2) has a unique corresponding $y$-value (1, 2, 3, 4).
  • This represents a valid function since every input has exactly one output.

Conclusion:

• Both b and d represent valid functions, but b is the first valid function according to the order in the options.

Thus, the correct answer is b.

Would you like more details or have any questions?

Here are 5 related questions to further your understanding:

1. What is the formal definition of a function?
2. Can a function have two identical $y$-values for different $x$-values?
3. How do we determine if a graph represents a function?
4. Can a function have multiple inputs that map to the same output?
5. How do vertical and horizontal lines relate to the concept of functions?

Tip: A quick way to test if a table represents a function is to check if any input $x$ has multiple outputs $y$. If it does, it’s not a function.