To determine whether the equation $x^2 + y^2 = 81$ defines $y$ as a function of $x$, we need to analyze the relationship.

Explanation:

1. A relationship defines $y$ as a function of $x$ if, for every value of $x$, there is exactly one corresponding value of $y$.

2. Rearrange the equation to solve for $y$: $y^2 = 81 - x^2$ $y = \pm \sqrt{81 - x^2}$ Here, $y$ can take two values (positive and negative square root) for most values of $x$ in the domain $-9 \leq x \leq 9$ (since $x^2 \leq 81$).

3. Because there are two possible values of $y$ for a single $x$ (except at $x = \pm 9$), this equation does not define $y$ as a function of $x$.

Answer:

No, the equation $x^2 + y^2 = 81$ does not define $y$ as a function of $x$.

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Related Questions:

1. What is the geometric interpretation of the equation $x^2 + y^2 = 81$?
2. How do we formally define a function in mathematics?
3. What would happen if we restricted $y$ to non-negative values in $x^2 + y^2 = 81$?
4. Can this equation define $x$ as a function of $y$?
5. What is the domain and range of the equation $x^2 + y^2 = 81$?

Tip:

Always use the vertical line test on a graph to quickly determine if a relationship is a function. If a vertical line intersects the graph at more than one point, it is not a function!