To determine which of the given options is not a function, we need to recall the definition of a function in the context of sets of ordered pairs. A set of ordered pairs is a function if, for every $x$ in the domain, there is exactly one $y$ in the codomain corresponding to $x$. This means that each input ($x$ value) must map to only one output ($y$ value). Let's analyze each option based on this criterion.

Option A: $$\left\{(x, y): x, y \in \mathbb{R}, x^2=y\right\}$$

This is a function because for each $x$ value, there is exactly one $y$ value such that $y = x^2$. This relationship passes the vertical line test, as no vertical line can intersect the graph at more than one point. Thus, Option A is indeed a function.

Option B: $$\left\{(x, y): x, y \in \mathbb{R}, y^2=x\right\}$$

This option is not a function. The reason is that for a given $x$ (except for $x=0$), there can be two different $y$ values such that $y^2=x$: one positive and one negative. For example, if $x=4$, both $y=2$ and $y=-2$ satisfy the equation $y^2=x$. This violates the definition of a function since one $x$ value corresponds to more than one $y$ value. Therefore, Option B is not a function.

Option C: $$\left\{(x, y): x, y \in \mathbb{R}, x=y^3\right\}$$

This is a function because for each $y$ value, there is exactly one $x$ value that can be calculated using $x = y^3$. No matter what $y$ value you choose, plugging it into $y^3$ will yield one unique $x$, making it a function.

Option D: $$\left\{(x, y): x, y \in \mathbb{R}, y=x^3\right\}$$

This also describes a function. For every $x$ value, $y = x^3$ produces a single $y$ value. This equation satisfies the function criterion as each input ($x$) has exactly one output ($y$).

Therefore, the correct answer to which of the following is not a function is Option B:

$$\left\{(x, y): x, y \in \mathbb{R}, y^2=x\right\}$$

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Functions · Mathematics · VITEEE

MCQ (Single Correct Answer)