To find the range of the function \( f(x) = x^2 + 2 \), we need to understand how the function behaves as \(x\) takes on different values.

### Step-by-step solution:
1. The term \( x^2 \) represents a parabola that opens upwards, and the smallest value of \( x^2 \) is 0 (which occurs when \( x = 0 \)).
   
2. Therefore, the minimum value of \( f(x) \) occurs when \( x = 0 \):
   \[
   f(0) = 0^2 + 2 = 2
   \]

3. As \( x^2 \) increases for values of \( x \) away from 0 (positive or negative), the function value \( f(x) \) will increase without bound. Thus, \( f(x) \) can grow infinitely large.

4. Hence, the range of \( f(x) = x^2 + 2 \) is all values of \( f(x) \) that are greater than or equal to 2.

### Conclusion:
The range of \( f(x) = x^2 + 2 \) is:
\[
[2, \infty)
\]

Would you like more details or have any questions?

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Here are five related questions:
1. What is the domain of \( f(x) = x^2 + 2 \)?
2. How would the range change if the function was \( f(x) = -x^2 + 2 \)?
3. What is the vertex of the parabola represented by \( f(x) = x^2 + 2 \)?
4. How does the range of \( f(x) = x^2 - 2 \) compare to this function?
5. How do you find the axis of symmetry for quadratic functions?

**Tip:** The range of quadratic functions is directly related to the direction the parabola opens (upward or downward) and its vertex.

Explore how to find the range of the quadratic function f(x) = x^2 + 2. This detailed solution covers key concepts such as the behavior of quadratic functions, the minimum value, and range calculation. Ideal for students in Grades 9-11.

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