Find the domain and range of the following functions:
`(x^(2))/(1+x^(2))`

Answer

To find the domain and range of the function \( f(x) = \frac{x^2}{1 + x^2} \), we will go through the following steps:

### Step 1: Determine the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined.

1. **Identify the function**:
\[ f(x) = \frac{x^2}{1 + x^2} \]

2. **Check for restrictions**:
The denominator \( 1 + x^2 \) cannot be zero. However, since \( x^2 \) is always non-negative, \( 1 + x^2 \) is always greater than or equal to 1. Therefore, it can never be zero.

3. **Conclusion about the domain**:
Since there are no restrictions on \( x \), the domain is:
\[ \text{Domain} = \{ x \in \mathbb{R} \} \quad \text{or} \quad (-\infty, \infty) \]

### Step 2: Determine the Range
The range of a function is the set of all possible output values (y-values) that the function can produce.

1. **Set the function equal to y**:
\[ y = \frac{x^2}{1 + x^2} \]

2. **Cross-multiply to eliminate the fraction**:
\[ y(1 + x^2) = x^2 \]
This simplifies to:
\[ y + yx^2 = x^2 \]

3. **Rearrange the equation**:
\[ yx^2 - x^2 + y = 0 \]
Factor out \( x^2 \):
\[ x^2(y - 1) + y = 0 \]

4. **Identify this as a quadratic equation in terms of \( x^2 \)**:
\[ x^2(y - 1) + y = 0 \]
For \( x^2 \) to be real, the discriminant must be greater than or equal to zero.

5. **Calculate the discriminant**:
The discriminant \( D \) of the quadratic \( ax^2 + bx + c = 0 \) is given by \( D = b^2 - 4ac \). Here, \( a = (y - 1) \), \( b = 0 \), and \( c = y \):
\[ D = 0^2 - 4(y - 1)(y) = -4y^2 + 4y \]
Set the discriminant \( D \geq 0 \):
\[ -4y^2 + 4y \geq 0 \]
This simplifies to:
\[ 4y(1 - y) \geq 0 \]

6. **Find the critical points**:
The critical points are \( y = 0 \) and \( y = 1 \).

7. **Test intervals**:
- For \( y < 0 \): \( 4y(1 - y) > 0 \) (positive)
- For \( 0 \leq y \leq 1 \): \( 4y(1 - y) \leq 0 \) (negative)
- For \( y > 1 \): \( 4y(1 - y) < 0 \) (negative)

8. **Conclusion about the range**:
The values of \( y \) that satisfy \( 4y(1 - y) \geq 0 \) are:
\[ \text{Range} = [0, 1) \]

### Final Result
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( [0, 1) \)