Which of the following is a function ?

Options

A. `{(x, y) : y = |x|, x, y in R}`
B. `{(x,y) : y^(2) = x , x, y in R}`
C. `{(x, y) : x^(2) + y^(2) = 1, x, y in R}`
D. `{(x, y) : x^(2) - y^(2) = 1, x, y in R}`

Answer

To determine which of the given options is a function, we will analyze each option based on the definition of a function and the graphical representation of each equation.

### Step-by-Step Solution:

1. **Understand the Definition of a Function**:
A relation is a function if every input (x-value) has exactly one output (y-value). This means that for any given x, there should be only one corresponding y. Additionally, a function can be one-to-one (injective) and onto (surjective), but for this question, we primarily need to check if each x has a unique y.

2. **Analyze Option 1: \( y = |x| \)**:
- The graph of \( y = |x| \) is a V-shaped graph that opens upwards.
- For every x-value, there is exactly one y-value.
- Therefore, this relation satisfies the definition of a function.

3. **Analyze Option 2: \( y^2 = x \)**:
- The graph of \( y^2 = x \) is a parabola that opens to the right.
- This graph is symmetric about the x-axis, meaning for a positive x-value, there are two corresponding y-values (one positive and one negative).
- Therefore, this relation does not satisfy the definition of a function.

4. **Analyze Option 3: \( x^2 + y^2 = 1 \)**:
- The graph of \( x^2 + y^2 = 1 \) represents a circle centered at the origin with a radius of 1.
- For any x-value in the range (-1, 1), there are two corresponding y-values (one above and one below the x-axis).
- Therefore, this relation does not satisfy the definition of a function.

5. **Analyze Option 4: \( x^2 - y^2 = 1 \)**:
- The graph of \( x^2 - y^2 = 1 \) represents a hyperbola.
- Similar to the previous options, for certain x-values, there are two corresponding y-values.
- Therefore, this relation does not satisfy the definition of a function.

6. **Conclusion**:
- Among the four options, only option 1, \( y = |x| \), is a function. Options 2, 3, and 4 do not satisfy the definition of a function.

### Final Answer:
**Option 1: \( y = |x| \) is a function.**

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