Consider the function `f: R to R` defined as `f(x)=(x)/(x^(2)+1)`
Assertion (A): f(x) is not one-one.
Reason (R): f(x) is not onto.

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is True

Answer

To solve the problem, we need to analyze the function \( f(x) = \frac{x}{x^2 + 1} \) and determine whether it is one-one (injective) and onto (surjective).

### Step 1: Check if \( f(x) \) is one-one

To check if the function is one-one, we can differentiate \( f(x) \) and analyze the sign of the derivative.

1. **Differentiate \( f(x) \)**:
\[ f'(x) = \frac{(x^2 + 1)(1) - x(2x)}{(x^2 + 1)^2} = \frac{x^2 + 1 - 2x^2}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2} \]

2. **Analyze the sign of \( f'(x) \)**:
- The denominator \( (x^2 + 1)^2 \) is always positive.
- The numerator \( 1 - x^2 \) changes sign at \( x = -1 \) and \( x = 1 \).
- Therefore, \( f'(x) > 0 \) when \( -1 < x < 1 \) and \( f'(x) < 0 \) when \( x < -1 \) or \( x > 1 \).

3. **Conclusion**:
- Since \( f'(x) \) is positive in the interval \( (-1, 1) \) and negative outside this interval, the function is not strictly increasing or strictly decreasing over its entire domain.
- Thus, \( f(x) \) is **not one-one**.

### Step 2: Check if \( f(x) \) is onto

To check if the function is onto, we need to determine the range of \( f(x) \).

1. **Set \( y = f(x) \)**:
\[ y = \frac{x}{x^2 + 1} \]
Rearranging gives:
\[ y(x^2 + 1) = x \implies yx^2 - x + y = 0 \]

2. **Analyze the discriminant**:
For \( x \) to be real, the discriminant of this quadratic equation must be non-negative:
\[ D = (-1)^2 - 4(y)(y) = 1 - 4y^2 \geq 0 \]
This implies:
\[ 1 - 4y^2 \geq 0 \implies 4y^2 \leq 1 \implies y^2 \leq \frac{1}{4} \implies -\frac{1}{2} \leq y \leq \frac{1}{2} \]

3. **Conclusion**:
- The range of \( f(x) \) is \( \left[-\frac{1}{2}, \frac{1}{2}\right] \).
- Since the codomain is \( \mathbb{R} \) (all real numbers) and the range is limited to \( \left[-\frac{1}{2}, \frac{1}{2}\right] \), \( f(x) \) is **not onto**.

### Final Conclusion

- Assertion (A): \( f(x) \) is not one-one. **True**
- Reason (R): \( f(x) \) is not onto. **True**

However, the reason does not correctly explain the assertion. Therefore, both are true, but the reason is not the correct explanation for the assertion.