Question

Determine whether \(y\) is a function of \(x\). $$ x^{2}+y^{2}=4 $$

Short answer

No, \(y\) is not a function of \(x\) in the given equation.

Step-by-step solution

Write the given equation

The given equation is: \(x^{2}+y^{2}=4\)

Rearrange the equation to solve for \(y\)

Next, rearrange the equation to solve it for \(y\). Subtract \(x^{2}\) from both sides of the equation. The rearranged equation is: \(y^{2}=4-x^{2}\). Thereafter, take square root on both sides, its gives two solutions for \(y\), \(y=\sqrt{4-x^{2}}\) and \(y=-\sqrt{4-x^{2}}\)

Determine if \(y\) is a function of \(x\)

For a relationship to be a function, every x value must have exactly one corresponding y value. Here for every x value, we have two y values, one from \(y=\sqrt{4-x^{2}}\) and one from \(y=-\sqrt{4-x^{2}}\). Hence, \(y\) is not a function of \(x\) in this equation.