Determine whether each equation defines y as a function of x. 2x + y^2 = 6

Verified step by step guidance

1. Rewrite the given equation: $2x\; +\; y^2\; =\; 6$. To determine if y is a function of x, we need to check if for every value of x, there is exactly one corresponding value of y.
2. Isolate the $y^2$ term by subtracting $2x$ from both sides: $y^2\; =\; 6\; -\; 2x$.
3. Take the square root of both sides to solve for y. Remember that taking the square root introduces both a positive and a negative solution: $y\; =\; \backslash (\backslash pm\backslash )\backslash (\backslash sqrt\{6\; -\; 2x\}\backslash )$.
4. Analyze the result: The presence of the $\backslash (\backslash pm\backslash )$ symbol indicates that for a single value of x, there are two possible values of y (one positive and one negative).
5. Conclude that the equation does not define y as a function of x because it fails the vertical line test, meaning a single x-value corresponds to more than one y-value.

Key Concepts

Function Definition

A function is a relation where each input (x-value) corresponds to exactly one output (y-value). To determine if an equation defines y as a function of x, we must check if for every x, there is a unique y. This means that for any given x, there should not be multiple y-values that satisfy the equation.

Vertical Line Test

The vertical line test is a visual method used to determine if a curve represents a function. If any vertical line drawn through the graph intersects it at more than one point, the relation is not a function. This test is particularly useful for analyzing graphs but can also inform our understanding of equations in implicit forms.

Solving for y

To analyze whether an equation defines y as a function of x, we often need to solve the equation for y. This involves isolating y on one side of the equation. If the resulting expression for y can yield multiple values for a single x, then y does not qualify as a function of x.