The given set $R_1 = \{(x, y) : x, y \in \mathbb{R} \text{ and } x^2 + y^2 = 1\}$ represents the set of all points in the plane $(x, y)$ that lie on a circle of radius 1 centered at the origin. Let’s analyze whether this is a function and why:

Definition of a Function

A relation is a function if every input $x$ has a unique output $y$. This means that for each $x \in \mathbb{R}$, there should be exactly one $y$ corresponding to it.

Analysis of $R_1$

The equation $x^2 + y^2 = 1$ is the equation of a circle. For a given $x$ (except at the extremes $x = -1$ or $x = 1$), there are two possible values of $y$: one positive and one negative, determined by: $y = \pm \sqrt{1 - x^2}.$ Thus, the same $x$ value produces two distinct $y$ values, violating the condition of uniqueness required for a function.

• For example, if $x = 0$, then $y = \pm 1$ (two values for $y$).
• If $x = \frac{\sqrt{2}}{2}$, then $y = \pm \frac{\sqrt{2}}{2}$ (again, two values for $y$).

Conclusion

The set $R_1$ does not represent a function because for some values of $x$, there are multiple corresponding values of $y$.

Key Points

• $R_1$ is a relation, not a function.
• It describes a circle, which fails the vertical line test for functions.

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5 Related Questions

1. What is the difference between a relation and a function in mathematics?
2. How does the vertical line test determine whether a relation is a function?
3. Can a relation like $R_1$ be restricted to make it a function?
4. How can you represent this circle parametrically instead of as a relation?
5. What types of relations always form valid functions?

Tip

When analyzing if a relation is a function, always check whether each input produces a unique output.