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Write F if the given rule defines Y as function of X and write R if it is not. Justify the answer

To determine if a given rule defines \( y \) as a function of \( x \), we need to check if every input value of \( x \) has exactly one corresponding output value of \( y \). If each \( x \) value is associated with exactly one \( y \) value, the relation is a function. If any \( x \) value corresponds to more than one \( y \) value, then the relation is not a function.

### Analysis of the Table:
From the table provided in the image:

\[
\begin{array}{c|c}
x & y \\
\hline
9 & 2 \\
4 & 3 \\
1 & 1 \\
0 & 0 \\
4 & -3 \\
9 & -3 \\
\end{array}
\]

- \( x = 9 \) corresponds to \( y = 2 \) and \( y = -3 \).  
- \( x = 4 \) corresponds to \( y = 3 \) and \( y = -3 \).

### Conclusion:
Since the \( x \) values 9 and 4 each correspond to two different \( y \) values, the given rule **does not** define \( y \) as a function of \( x \). Therefore, the correct response is **R** (it is not a function).

If you have any more questions or need further details, feel free to ask!

### Related Questions:
1. What is the definition of a function in terms of input and output?
2. How can you determine whether a relation is a function by using a graph?
3. What is the vertical line test, and how does it help in identifying functions?
4. Can a function have the same \( y \) value for different \( x \) values? Explain.
5. What are some common types of functions (e.g., linear, quadratic) and their characteristics?

### Tip:
When determining whether a relation is a function, always check if each input has a unique output. If any input has multiple outputs, the relation is not a function.

Understand how to determine if a given rule defines y as a function of x. Analyze a specific table to identify whether each x value has a unique y value, crucial for understanding functions in mathematics.

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