Determine whether each equation defines y as a function of x. x = (1/3)(y²)

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Start by examining the given equation: $x = \frac{1}{3} y^{2}$. This equation relates $x$ and $y$.

To determine if $y$ is a function of $x$, try to express $y$ explicitly in terms of $x$. Multiply both sides by 3 to isolate $y^{2}$: $3x = y^{2}$.

Next, solve for $y$ by taking the square root of both sides: $y = \pm \sqrt{3x}$. Notice the $\pm$ sign indicates two possible values of $y$ for each $x$ (except when $x=0$).

Recall the definition of a function: for each input $x$, there must be exactly one output $y$. Since here each $x$ (greater than or equal to zero) corresponds to two $y$ values, $y$ is not a function of $x$.

Therefore, the equation does not define $y$ as a function of $x$ because it fails the vertical line test and does not assign a unique $y$ to each $x$.

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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). To determine if an equation defines y as a function of x, we check if for every x there is only one y.

Solving for y in Terms of x

To analyze if y is a function of x, we often solve the equation for y explicitly. This helps identify if multiple y-values correspond to a single x-value, which would mean y is not a function of x.

Vertical Line Test and Relation Symmetry

The vertical line test visually checks if a curve represents y as a function of x by ensuring vertical lines intersect the graph at most once. Equations involving y² often produce symmetric graphs, indicating multiple y-values for one x.

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