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Can the equation 𝑥² + 𝑦² = 4 be expressed as a function? If yes, state the function.

Can the equation 𝑥 squared plus 𝑦 squared equals four be expressed as a function? If yes, state the function.

We recall that the vertical line test can be used to determine whether a graph represents a function. This means that in order to establish whether the equation 𝑥 squared plus 𝑦 squared equals four can be expressed as a function, we can sketch its graph. Any equation in the form 𝑥 squared plus 𝑦 squared equals 𝑟 squared represents a circle which is centered at the origin with radius 𝑟. In this question, as 𝑟 squared is equal to four and 𝑟 must be positive, the radius of our circle is two. Sketching the circle, we see that it intersects the 𝑥- and 𝑦-axis at two and negative two.

Adding a vertical line to our diagram, we note that it intersects the circle twice. As there is more than one point of intersection, the graph does not represent a function. We can therefore conclude that the equation 𝑥 squared plus 𝑦 squared equals four cannot be expressed as a function. The correct answer is no.

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