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The question asks to find the range of the given functions. Let's go through each of them:

1. \( x^2 - 6x + 8 \)
   - This is a quadratic function of the form \( f(x) = ax^2 + bx + c \). The range of a quadratic function depends on whether it opens upwards or downwards and its vertex.

2. \( x^2 + 8x \)
   - Another quadratic function. It will follow a similar analysis, determining the vertex and direction of the parabola.

3. \( x^2 - 2 \)
   - This is again a quadratic function, where the range can be found by determining the vertex and the behavior of the parabola.

Let me solve each part:

### a) \( f(x) = x^2 - 6x + 8 \)
To determine the range:
1. Complete the square:
   \[
   f(x) = x^2 - 6x + 8
   \]
   Factor it:
   \[
   f(x) = (x - 3)^2 - 1
   \]
   The vertex form of a quadratic function is \( a(x-h)^2 + k \). Here, the vertex is \( (3, -1) \), and since the coefficient of \( (x-3)^2 \) is positive, the parabola opens upwards.

Thus, the minimum value of the function is \( -1 \), and the range is \( [-1, \infty) \).

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Do you need details for the next parts? Let me know if you'd like me to continue solving them!

Here are five related questions:
1. How do you complete the square for quadratic functions?
2. What is the general formula for the range of a quadratic function?
3. What does the vertex of a quadratic function tell you?
4. How does the sign of the quadratic term affect the graph's direction?
5. What is the significance of the discriminant in quadratic functions?

**Tip:** When analyzing quadratic functions, converting them to vertex form simplifies determining the range.

Help me with finding the range and functions.

Discover how to find the range of quadratic functions including x^2 - 6x + 8, x^2 + 8x, and x^2 - 2. This guide provides step-by-step methods using vertex and standard forms, ideal for high school students learning algebra.

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