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

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Recall the definition of a function: for each input value of x, there must be exactly one output value of y.

Look at the given equation: $y = 6 - x^{2}$.

For any value of x, calculate the right side $6 - x^{2}$ to find the corresponding y value.

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.

Therefore, the equation defines y as a function of x because each x corresponds to exactly one y.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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