The equation $x^2 + y^2 = 16$ represents a circle with a radius of 4 centered at the origin in the coordinate plane.

To determine whether this equation defines $y$ as a function of $x$, we can try to solve for $y$.

Starting with:

$x^2 + y^2 = 16$

Isolate $y^2$:

$y^2 = 16 - x^2$

Now, take the square root of both sides:

$y = \pm \sqrt{16 - x^2}$

This results in two values for $y$ (positive and negative) for each $x$ in the range $[-4, 4]$. This means that for most $x$-values, there are two possible corresponding $y$-values, which violates the definition of a function. A function can only have one output for each input.

Conclusion: No, the equation $x^2 + y^2 = 16$ does not define $y$ as a function of $x$, because for many values of $x$, there are two corresponding values of $y$.

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Would you like to dive deeper into the graph of the equation or explore other related concepts?

Related Questions:

1. How do implicit functions relate to equations like $x^2 + y^2 = 16$?
2. What are some conditions that make an equation define $y$ as a function of $x$?
3. How can you find the domain and range for the equation $x^2 + y^2 = 16$?
4. What is the difference between a circle equation and an equation of a function?
5. How can you use parametric equations to describe the circle $x^2 + y^2 = 16$?

Tip:

When analyzing equations like this, always check for multiple values of $y$ that might correspond to a single $x$. If there are multiple $y$-values, it's not a function.