Questions: Determine whether the equation is a function. x + y^2 = 1 Select the correct answer below: The equation is a function. The equation is not a function.

Solution Steps

To determine whether the equation \( x + y^2 = 1 \) is a function, we need to check if for every value of \( x \), there is exactly one corresponding value of \( y \). If there are multiple values of \( y \) for a single \( x \), then it is not a function.

We start with the equation: \[ x + y^2 = 1 \]

To determine if this equation represents a function, we solve for \( y \): \[ y^2 = 1 - x \] \[ y = \pm \sqrt{1 - x} \]

For a given value of \( x \), there are generally two solutions for \( y \): \[ y = \sqrt{1 - x} \quad \text{and} \quad y = -\sqrt{1 - x} \]

For example, if \( x = 0 \): \[ y = \sqrt{1 - 0} = 1 \quad \text{and} \quad y = -\sqrt{1 - 0} = -1 \] Thus, for \( x = 0 \), there are two values of \( y \): \( y = 1 \) and \( y = -1 \).

A function must have exactly one output for each input. Since there are two values of \( y \) for a single value of \( x \), the equation \( x + y^2 = 1 \) does not satisfy the definition of a function.

Final Answer