find the domain and range of the function
` f : R to R : f (x) =x^(2) +1`

Explanation

To find the domain and range of the function \( f(x) = x^2 + 1 \), we will follow these 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.

For the function \( f(x) = x^2 + 1 \):
- The expression \( x^2 \) is defined for all real numbers.
- Adding 1 does not restrict the input values.

Thus, the domain of \( f \) is:
\[ \text{Domain} = \{ x \in \mathbb{R} \} \]
or simply:
\[ \text{Domain} = \mathbb{R} \]

### 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.

For \( f(x) = x^2 + 1 \):
- The term \( x^2 \) is always non-negative (i.e., \( x^2 \geq 0 \)).
- Therefore, the smallest value of \( x^2 \) is 0, which occurs when \( x = 0 \).
- Consequently, the smallest value of \( f(x) \) is:
\[ f(0) = 0 + 1 = 1 \]
- As \( x \) increases or decreases from 0, \( x^2 \) increases, and thus \( f(x) \) will also increase without bound.

Therefore, the range of \( f \) is:
\[ \text{Range} = \{ y \in \mathbb{R} : y \geq 1 \} \]

### Final Result
- **Domain**: \( \mathbb{R} \)
- **Range**: \( [1, \infty) \)