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.