https://file.aiquickdraw.com/mathbot/file/1726928693632vnu0ft8z.jpg

Write F if function and NF if not function in a given problem.

The equation provided is:

\[
y^2 + x^2 = 9
\]

This represents the equation of a circle with a radius of 3, centered at the origin \((0, 0)\).

To determine whether it defines a function, we can use the **vertical line test**, which checks whether any vertical line crosses the graph more than once. For an equation to represent a function, each value of \( x \) should correspond to exactly one value of \( y \).

For the equation of a circle, when solving for \( y \), we get:

\[
y^2 = 9 - x^2
\]
\[
y = \pm \sqrt{9 - x^2}
\]

This shows that for a given \( x \), there are generally two corresponding values of \( y \) (one positive and one negative). Since one value of \( x \) can give two values of \( y \), the equation does **not** represent a function.

### Conclusion:
The equation is **NF (not a function)**.

Would you like any more details or have any further questions? Here are some related questions for practice:

1. How would you graph the equation of this circle?
2. How can you convert this equation to parametric form?
3. What would the equation look like if it were a function?
4. How would this equation change if the radius were doubled?
5. How do you define a function algebraically?

**Tip:** A function must pass the vertical line test, meaning no vertical line should intersect the graph at more than one point.

Determine whether the equation y^2 + x^2 = 9 represents a function. Learn about the equation of a circle, the vertical line test, and why this equation is not classified as a function. Suitable for students in Grades 9-12.

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