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Let's go step-by-step and solve the two parts:

### 1. Finding \( f(x + 8) \)
Given:
\[ f(x) = x^2 - 2x - 6 \]

To find \( f(x + 8) \), substitute \( x + 8 \) into the function \( f(x) \):
\[
f(x + 8) = (x + 8)^2 - 2(x + 8) - 6
\]

Expand the terms:

- \( (x + 8)^2 = x^2 + 16x + 64 \)
- \( -2(x + 8) = -2x - 16 \)

Now, substitute these back into the expression for \( f(x + 8) \):
\[
f(x + 8) = x^2 + 16x + 64 - 2x - 16 - 6
\]
Simplify:
\[
f(x + 8) = x^2 + 14x + 42
\]

So, 
\[
f(x + 8) = x^2 + 14x + 42
\]

### 2. Finding \( g\left( \frac{x}{4} \right) \)
Given:
\[ g(x) = \frac{5x - 1}{9x - 4} \]

To find \( g\left( \frac{x}{4} \right) \), substitute \( \frac{x}{4} \) into the function \( g(x) \):
\[
g\left( \frac{x}{4} \right) = \frac{5\left( \frac{x}{4} \right) - 1}{9\left( \frac{x}{4} \right) - 4}
\]

Simplify the numerator and denominator:
- Numerator: \( 5 \times \frac{x}{4} = \frac{5x}{4} \), so the numerator becomes \( \frac{5x}{4} - 1 \)
- Denominator: \( 9 \times \frac{x}{4} = \frac{9x}{4} \), so the denominator becomes \( \frac{9x}{4} - 4 \)

Now combine the terms:
\[
g\left( \frac{x}{4} \right) = \frac{\frac{5x}{4} - 1}{\frac{9x}{4} - 4}
\]

To eliminate the fractions, multiply both the numerator and the denominator by 4:
\[
g\left( \frac{x}{4} \right) = \frac{5x - 4}{9x - 16}
\]

So,
\[
g\left( \frac{x}{4} \right) = \frac{5x - 4}{9x - 16}
\]

### Final Answers:
1. \( f(x + 8) = x^2 + 14x + 42 \)
2. \( g\left( \frac{x}{4} \right) = \frac{5x - 4}{9x - 16} \)

Would you like a detailed explanation of any part of this process or have further questions?

Here are 5 related questions:
1. How do you evaluate functions with more complex expressions like \( f(x - 2) \)?
2. What happens if we evaluate \( g(2x) \) for the given function \( g(x) \)?
3. Can you explain how to simplify algebraic fractions like \( \frac{5x}{4} - 1 \)?
4. What if you needed to compute \( f(x^2) \)? How would that change the approach?
5. How do you handle cases where the substitution leads to undefined values, such as \( g(x) \) when the denominator becomes zero?

**Tip:** Always simplify the expression fully after substitution to avoid missing hidden terms that might cancel out or combine.

The functions f and g are defined as follows: f(x) = x^2 - 2x - 6 and g(x) = (5x - 1) / (9x - 4). Find f(x + 8) and g(x/4). Write your answers without parentheses and simplify them as much as possible.

Explore how to substitute and simplify expressions for the functions f(x) = x^2 - 2x - 6 and g(x) = (5x - 1) / (9x - 4). Learn step-by-step how to find f(x + 8) and g(x/4) with detailed explanations. Suitable for high school students in Grades 10-12.

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