A real valued function is given by `f(x)=x^(2)+4`, find its domain and range.

Answer

To find the domain and range of the function \( f(x) = x^2 + 4 \), we will proceed step by step.

### Step 1: Identify the function
The given function is:
\[ f(x) = x^2 + 4 \]

### Step 2: Determine the domain
The domain of a function refers to the set of all possible input values (x-values) that can be used in the function.

- Since \( x^2 \) is defined for all real numbers (you can square any real number), we can conclude that:
\[ \text{Domain} = \{ x \in \mathbb{R} \} \]
This means that \( x \) can take any real number value.

### Step 3: Determine the range
The range of a function refers to the set of all possible output values (f(x)-values) that the function can produce.

- Let's analyze \( f(x) = x^2 + 4 \):
- The term \( x^2 \) is always non-negative (i.e., \( x^2 \geq 0 \) for all \( x \)).
- Therefore, the smallest value of \( x^2 \) is 0, which occurs when \( x = 0 \).
- Substituting \( x = 0 \) into the function gives:
\[ f(0) = 0^2 + 4 = 4 \]
- As \( x \) increases or decreases from 0, \( x^2 \) increases, which means \( f(x) \) will also increase. Thus, \( f(x) \) will always be greater than or equal to 4.

- Therefore, the range of \( f(x) \) is:
\[ \text{Range} = [4, \infty) \]

### Final Answer
- **Domain**: \( x \in \mathbb{R} \)
- **Range**: \( f(x) \in [4, \infty) \)

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