Examine whether the following functions are one-one, many-one or into, onto `f(x)=1/(1+x^(2)),(-oo,0)toR`

Explanation

To determine whether the function \( f(x) = \frac{1}{1 + x^2} \) defined from the interval \( (-\infty, 0) \) to \( \mathbb{R} \) is one-one, many-one, into, or onto, we will follow these steps:

### Step 1: Check if the function is one-one or many-one
To check if the function is one-one, we can differentiate it and analyze the derivative.

1. **Differentiate the function**:
\[ f(x) = \frac{1}{1 + x^2} \]
We will use the quotient rule to differentiate \( f(x) \):
- Let \( u = 1 \) and \( v = 1 + x^2 \).
- Then, \( u' = 0 \) and \( v' = 2x \).

Using the quotient rule:
\[ f'(x) = \frac{u'v - uv'}{v^2} = \frac{0 \cdot (1 + x^2) - 1 \cdot (2x)}{(1 + x^2)^2} = \frac{-2x}{(1 + x^2)^2} \]

2. **Set the derivative equal to zero**:
\[ -2x = 0 \implies x = 0 \]
However, \( x = 0 \) is not in the domain \( (-\infty, 0) \). Therefore, there are no critical points in the interval.

3. **Analyze the sign of the derivative**:
- For \( x < 0 \), \( -2x > 0 \), which means \( f'(x) > 0 \).
- Since \( f'(x) > 0 \) for all \( x \) in the domain, the function is strictly increasing.

Since the function is strictly increasing in the interval \( (-\infty, 0) \), it is one-one.

### Step 2: Check if the function is into or onto
Next, we need to determine the range of the function to see if it is into or onto.

1. **Find the range of the function**:
- As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( 0 \).
- At \( x = 0 \) (not included in the domain), \( f(0) = 1 \).
- Therefore, as \( x \) varies from \( -\infty \) to \( 0 \), \( f(x) \) takes values from \( 0 \) to \( 1 \) (not including \( 0 \) and including \( 1 \)).

Thus, the range of \( f(x) \) is \( (0, 1] \).

2. **Compare the range with the codomain**:
- The codomain is \( \mathbb{R} \).
- The range \( (0, 1] \) does not cover all of \( \mathbb{R} \).

Since the range is not equal to the codomain, the function is into.

### Conclusion
- The function \( f(x) = \frac{1}{1 + x^2} \) is **one-one** and **into**.