To solve for \( a \) in the equation \( f(4a) = 6f(a - 4) \), let's proceed step by step.

1. We aren't given an explicit form of the function \( f \), so without additional information, we can't directly simplify the equation. However, we can explore different potential cases based on common types of functions, such as assuming \( f(x) \) is a linear or constant function, or see if there are other reasonable assumptions.

### Case 1: Assume \( f(x) \) is a constant function
If \( f(x) \) is constant, say \( f(x) = c \), then the equation becomes:
\[
f(4a) = 6f(a - 4) \implies c = 6c.
\]
This implies:
\[
c = 0.
\]
Thus, if \( f(x) \) is constant, the solution is \( f(x) = 0 \) for all \( x \).

### Case 2: Assume \( f(x) = x \) (a linear function)
If \( f(x) = x \), then:
\[
f(4a) = 4a \quad \text{and} \quad f(a - 4) = a - 4.
\]
Substituting these into the original equation:
\[
4a = 6(a - 4).
\]
Now, solve for \( a \):
\[
4a = 6(a - 4) \implies 4a = 6a - 24.
\]
Simplifying:
\[
4a - 6a = -24 \implies -2a = -24 \implies a = 12.
\]

### Conclusion:
If \( f(x) \) is a linear function like \( f(x) = x \), the solution to the equation is \( a = 12 \).

Do you want further details, or have any questions about this? 

Here are five related questions to deepen understanding:
1. What if \( f(x) \) were a quadratic function? How would the solution change?
2. Can we try \( f(x) = x^2 \) and see how it affects the outcome?
3. What methods could we use if \( f(x) \) were exponential?
4. Why is assuming \( f(x) \) constant the simplest case?
5. How does the assumption of linearity simplify the problem?

**Tip:** When solving functional equations, making reasonable assumptions about the nature of \( f(x) \) can significantly simplify the problem.

Explore how to solve the functional equation f(4a) = 6f(a - 4) by assuming different types of functions, such as constant or linear. This problem highlights key mathematical concepts, including linearity and properties of functions. Suitable for students in Grades 10-12.

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