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To solve the given problem, we will proceed as follows:

### (i) Find \( f(-1) \) and \( f(3) \)
The function given is \( f(x) = x^2 - 10x + 16 \).

1. **Finding \( f(-1) \)**:
   \[
   f(-1) = (-1)^2 - 10(-1) + 16 = 1 + 10 + 16 = 27
   \]

2. **Finding \( f(3) \)**:
   \[
   f(3) = (3)^2 - 10(3) + 16 = 9 - 30 + 16 = -5
   \]

### (ii) State the range of \( f(x) \)

To find the range of \( f(x) = x^2 - 10x + 16 \) for \( 2 < x < 8 \), we will:

1. **Complete the square** to find the vertex:
   \[
   f(x) = x^2 - 10x + 16 = (x - 5)^2 - 25 + 16 = (x - 5)^2 - 9
   \]
   The vertex of the parabola \( (x - 5)^2 - 9 \) is at \( x = 5 \) and \( f(5) = -9 \).

2. **Evaluate \( f(x) \) at the endpoints of the interval \( 2 < x < 8 \)**:
   - At \( x = 2 \):
     \[
     f(2) = 2^2 - 10(2) + 16 = 4 - 20 + 16 = 0
     \]
   - At \( x = 8 \):
     \[
     f(8) = 8^2 - 10(8) + 16 = 64 - 80 + 16 = 0
     \]

Since the parabola opens upwards, the minimum value within \( 2 < x < 8 \) occurs at the vertex \( x = 5 \), and we have evaluated the function at the endpoints of the interval to find the maximum values. Therefore, the range of \( f(x) \) for \( 2 < x < 8 \) is:
\[
[-9, 0]
\]

### Summary
(i) \( f(-1) = 27 \) and \( f(3) = -5 \)

(ii) The range of \( f(x) \) for \( 2 < x < 8 \) is \([-9, 0]\).

Would you like more details on any of these steps, or do you have any questions?

### Related Questions
1. What is the vertex form of a quadratic function?
2. How do you complete the square for a quadratic equation?
3. How can you determine the range of a quadratic function on a given interval?
4. What are the properties of parabolas that open upwards?
5. How do you find the maximum and minimum values of a function on a closed interval?
6. What is the significance of the vertex in a parabola?
7. How do transformations affect the graph of a quadratic function?
8. Can you explain how to solve quadratic equations by factoring?

### Tip
When finding the range of a quadratic function within a specific interval, always evaluate the function at the endpoints and the vertex if it lies within the interval.

Learn how to find the values of f(-1) and f(3), and determine the range of the quadratic function f(x) = x^2 - 10x + 16 on the interval (2, 8). This problem covers key concepts such as quadratic functions, vertex form, completing the square, and properties of parabolas. Suitable for high school students in Grades 10-12.

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