We are given the function $f(x) = x^2 + 1$ and its domain is restricted to $(-2, 4)$, meaning the function is defined for all $x$ in the interval $(-2, 4)$. We need to find the range of this function on this restricted domain.

Steps to find the range:

1. Analyze the function: The function is $f(x) = x^2 + 1$, which is a parabola opening upwards. The value of $f(x)$ depends on the value of $x^2$.

2. Determine the behavior of the function on the domain: The domain is $(-2, 4)$, so we evaluate the function at the endpoints of this interval.

   • When $x = -2$, $f(-2) = (-2)^2 + 1 = 4 + 1 = 5$.
   • When $x = 4$, $f(4) = (4)^2 + 1 = 16 + 1 = 17$.

   Since the function $f(x) = x^2 + 1$ is increasing in the interval $[0, \infty)$ and decreasing in the interval $(-\infty, 0]$, we expect the minimum value to occur at the endpoints or near the vertex.

3. Find the minimum and maximum values of the function:

   • The function is symmetric, so it achieves a minimum value at $x = 0$, which is the midpoint of the interval $(-2, 4)$.
   • At $x = 0$, $f(0) = 0^2 + 1 = 1$.

4. Range of the function: Since $f(x)$ increases from $f(0) = 1$ to $f(4) = 17$, and decreases from $f(0) = 1$ to $f(-2) = 5$, the range of $f$ on $(-2, 4)$ is the interval:

   $[1, 17)$

Conclusion:

The range of the function $f(x) = x^2 + 1$ when the domain is restricted to $(-2, 4)$ is $[1, 17)$.

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Would you like a more detailed explanation of any of the steps? Here are 5 related questions to explore further:

1. How would the range change if the domain was $[-2, 4]$ instead of $(-2, 4)$?
2. What happens to the range if the function was $f(x) = x^2 - 1$?
3. How would you find the range of the function if the domain was restricted to a different interval?
4. How can we determine whether a function is increasing or decreasing in an interval?
5. What is the vertex of the function $f(x) = x^2 + 1$, and how does it relate to the range?

Tip: Always check the endpoints and critical points when working with restricted domains to ensure you've covered all potential maximum and minimum values!