Determine whether each equation defines y as a function of x. y = 6 -x²

Verified step by step guidance

1. Recall the definition of a function: for each input value of x, there must be exactly one output value of y.
2. Look at the given equation: \(y = 6 - x^{2}\).
3. For any value of x, calculate the right side \$6 - x^{2}$ to find the corresponding y value.
4. Since squaring x (\(x^{2}\)) always gives a non-negative result and subtracting it from 6 gives a unique y for each x, there is only one y for each x.
5. Therefore, the equation defines y as a function of x because each x corresponds to exactly one y.

Key Concepts

Definition of a Function

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). This means for every x, there is only one y. Understanding this helps determine if an equation defines y as a function of x.

Evaluating Equations for Function Criteria

To check if an equation defines y as a function of x, substitute values of x and see if y has a unique value. If each x produces only one y, the equation represents a function.

Quadratic Functions and Their Graphs

The equation y = 6 - x² is a quadratic function, which graphs as a parabola opening downward. Since for each x there is only one y, quadratic equations like this define y as a function of x.