Which equation below defines y as a function of x?
Understand the Problem
The question is asking to identify which equation out of the given options defines y as a function of x. This involves understanding the criteria for a relation to define a function, which typically means that for each input x, there should be a unique output y.
Answer
The equation that defines $y$ as a function of $x$ is $3y - x^2 = 0$.
Answer for screen readers
The equation that defines $y$ as a function of $x$ is:
$$ 3y - x^2 = 0 $$
Steps to Solve
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Analyze each equation for function definition To determine which equation defines $y$ as a function of $x$, check if each input $x$ corresponds to exactly one output $y$.
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Evaluate the first equation: $3x - y^2 = 0$ Rearranging gives $y^2 = 3x$. This implies $y = \pm \sqrt{3x}$. Since this results in two different outputs for each positive $x$, it does not define $y$ as a function of $x$.
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Evaluate the second equation: $3y - x^2 = 0$ Rearranging gives $y = \frac{x^2}{3}$. For any $x$, this results in a unique $y$. Thus, this does define $y$ as a function of $x$.
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Evaluate the third equation: $x^2 + y^2 = 9$ This equation represents a circle with radius 3. For most $x$ values, there are two corresponding $y$ values, which means it does not define $y$ as a function of $x$.
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Evaluate the fourth equation: $\frac{x}{y} = y$ Rearranging gives $x = y^2$. This implies $y = \pm \sqrt{x}$, which means this does not define $y$ as a function of $x$.
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Evaluate the fifth equation: $y = \pm \sqrt{4 - x^2}$ This represents the upper and lower halves of a circle. For most $x$ values, there are two corresponding $y$ values, meaning it does not define a function.
The equation that defines $y$ as a function of $x$ is:
$$ 3y - x^2 = 0 $$
More Information
This equation shows that for every $x$ value, there is a single corresponding $y$ value, adhering to the definition of a function.
Tips
- A common mistake is assuming that equations involving squares (like $y = \pm \sqrt{...}$) define functions without considering the $\pm$ sign, which indicates multiple values.
- Not checking if an equation results in multiple outputs for a single input can lead to incorrectly identifying a function.
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