Let's solve the problem step by step.

The function is $f(x) = -4x + 4$, and the domain is $\{-6, -4, 8\}$. We need to find the range, which means calculating the output of $f(x)$ for each value in the domain.

Step 1: Evaluate $f(x)$ for $x = -6$

$f(-6) = -4(-6) + 4 = 24 + 4 = 28$

Step 2: Evaluate $f(x)$ for $x = -4$

$f(-4) = -4(-4) + 4 = 16 + 4 = 20$

Step 3: Evaluate $f(x)$ for $x = 8$

$f(8) = -4(8) + 4 = -32 + 4 = -28$

Step 4: Write the range in set notation

The outputs (range values) are $28, 20,$ and $-28$. In set notation, the range is: $\{28, 20, -28\}$

Final Answer:

The range of $f(x)$ is: $\boxed{\{28, 20, -28\}}$

Let me know if you have any questions or want additional details!

Expanding Questions:

1. What is the definition of a function's range, and how is it determined?
2. How would the range change if the domain included all real numbers?
3. What happens to the range if you change the function to $f(x) = 4x + 4$?
4. Can you explain how to verify the correctness of the range using a graph?
5. How do you handle functions with more complex domains (e.g., intervals)?

Tip:

When calculating the range of a function for a specific domain, always substitute each domain value into the function carefully and double-check your arithmetic.