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.