Does the equatation \( x^{2}+y^{2}=16 \) define \( y \) as a function of \( x \) ?

To determine whether the equation \( x^{2} + y^{2} = 16 \) defines \( y \) as a function of \( x \), we can analyze the equation. The equation \( x^{2} + y^{2} = 16 \) represents a circle centered at the origin (0, 0) with a radius of 4. To see if \( y \) can be expressed as a function of \( x \), we can try to solve for \( y \): \[ y^{2} = 16 - x^{2} \] Taking the square root of both sides gives us: \[ y = \sqrt{16 - x^{2}} \quad \text{and} \quad y = -\sqrt{16 - x^{2}} \] This means that for each value of \( x \) (within the interval \([-4, 4]\)), there are generally two corresponding values of \( y \) (one positive and one negative). Since a function is defined as a relation where each input (in this case, each \( x \)) corresponds to exactly one output (one \( y \)), the equation \( x^{2} + y^{2} = 16 \) does not define \( y \) as a function of \( x \) because there are two possible values of \( y \) for most values of \( x \). Thus, the answer is **no**, the equation \( x^{2} + y^{2} = 16 \) does not define \( y \) as a function of \( x \).