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Let's break down the problem from the image step by step.

1. **Determine if \( x = y + 8 \) represents a linear function:**

   The given equation is \( x = y + 8 \). We can rearrange this equation to express \( y \) in terms of \( x \):
   \[
   y = x - 8
   \]
   This is a linear equation because it is in the form \( y = mx + b \), where \( m \) (the slope) is 1, and \( b \) (the y-intercept) is \(-8\). Therefore, the equation \( x = y + 8 \) does represent a linear function.

2. **Determine algebraically if the equation is even, odd, or neither:**

   To check whether a function is even, odd, or neither, we express the function as \( f(y) = x = y + 8 \).

   - **For even functions**, the following condition must hold: \( f(-y) = f(y) \).
   - **For odd functions**, the condition must be: \( f(-y) = -f(y) \).

   Let's check:

   \[
   f(-y) = (-y) + 8 = -y + 8
   \]
   Comparing this to the original function \( f(y) = y + 8 \), we see that:
   \[
   f(-y) \neq f(y)
   \]
   and
   \[
   f(-y) \neq -f(y) \quad \text{(since} \, -f(y) = -(y + 8) = -y - 8\text{)}.
   \]
   Hence, the function is **neither even nor odd**.

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Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. What is the slope of the function \( y = x - 8 \)?
2. What is the graphical interpretation of an even function?
3. Can a function be both even and odd? Why or why not?
4. How do you graphically identify whether a function is even?
5. What is the domain and range of the function \( y = x - 8 \)?

**Tip:** When checking whether a function is even or odd, always try substituting \( -x \) and see how the function transforms!

Determine whether x = y + 8 represents a linear function and then algebraically determine whether the equation is even, odd, or neither.

Explore whether the equation x = y + 8 is a linear function and algebraically determine if it is even, odd, or neither. This problem involves key algebraic concepts like linear functions and symmetry checks. Suitable for high school students in grades 9-11.

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