If `f(x)=x^(2)+2`, which of the following could be a value of f(x)?

Options

A. `-1`
B. `0`
C. `1`
D. `2`

Answer

To solve the problem, we need to analyze the function \( f(x) = x^2 + 2 \) and determine which of the given options could be a value of \( f(x) \).

### Step-by-Step Solution:

1. **Understand the function**: The function given is \( f(x) = x^2 + 2 \). This is a quadratic function where \( x^2 \) represents the square of \( x \).

2. **Identify the minimum value of \( f(x) \)**: Since \( x^2 \) is always non-negative (i.e., \( x^2 \geq 0 \)), the smallest value \( x^2 \) can take is 0. Therefore, the minimum value of \( f(x) \) occurs when \( x^2 = 0 \):
\[ f(x) = 0 + 2 = 2 \]
This means \( f(x) \) is always greater than or equal to 2.

3. **Set up the inequality**: From the above analysis, we can conclude:
\[ f(x) \geq 2 \]
This tells us that the function \( f(x) \) can never take a value less than 2.

4. **Evaluate the options**:
- **Option 1**: -1
- Since -1 is less than 2, it cannot be a value of \( f(x) \).
- **Option 2**: 0
- Since 0 is also less than 2, it cannot be a value of \( f(x) \).
- **Option 3**: 1
- Since 1 is less than 2, it cannot be a value of \( f(x) \).
- **Option 4**: 2
- Since 2 is equal to the minimum value of \( f(x) \), it can be a value of \( f(x) \).

5. **Conclusion**: The only option that satisfies the condition \( f(x) \geq 2 \) is option 4, which is 2. Therefore, the correct answer is:
\[ \text{Option 4: } 2 \]