II. CHOOSE THE CORRECT ANSWER FROM THE GIVEN ALTERNATIVES. 3. Which of the following relation is a function? A. \( R=\left\{(x, y): x, y \in \mathbb{R}\right. \) and \( \left.x^{2}+y^{2}=1\right\} \) B. \( R=\left\{(x, y): x, y \in \mathbb{R}\right. \) and \( \left.x=y^{2}\right\} \) C. \( R=\{(x, y): x, y \in \mathbb{R} \) and \( y \geq x+1\} \quad \) D. \( R=\{(x, y): x, y \in \mathbb{R} \) and \( |x|+2=y\} \)

To determine which of the given relations is a function, we need to verify if each relation assigns **exactly one** \( y \) value to each \( x \) value in its domain. Let's analyze each option: ### **A. \( R=\left\{(x, y): x, y \in \mathbb{R} \text{ and } x^{2}+y^{2}=1\right\} \)** - **Interpretation**: This represents the unit circle. - **Analysis**: For most \( x \) values in the interval \([-1, 1]\), there are **two** corresponding \( y \) values (positive and negative). - **Conclusion**: **Not a function** because a single \( x \) maps to multiple \( y \) values. ### **B. \( R=\left\{(x, y): x, y \in \mathbb{R} \text{ and } x=y^{2}\right\} \)** - **Interpretation**: This represents a parabola. - **Analysis**: For \( x > 0 \), there are **two** corresponding \( y \) values (positive and negative). - **Conclusion**: **Not a function** because a single \( x \) maps to multiple \( y \) values. ### **C. \( R=\{(x, y): x, y \in \mathbb{R} \text{ and } y \geq x+1\} \)** - **Interpretation**: This represents a region above the line \( y = x + 1 \). - **Analysis**: For each \( x \), there are **infinitely many** \( y \) values that satisfy the inequality. - **Conclusion**: **Not a function** because a single \( x \) maps to multiple \( y \) values. ### **D. \( R=\{(x, y): x, y \in \mathbb{R} \text{ and } |x|+2=y\} \)** - **Interpretation**: This represents the function \( y = |x| + 2 \). - **Analysis**: For each \( x \), there is **exactly one** \( y \) value. - **Conclusion**: **This is a function** because each \( x \) maps to exactly one \( y \). **Final Answer:** **Option D.** \( R=\{(x, y): x, y \in \mathbb{R} \) and \( |x|+2=y\} \)