To evaluate the piecewise function $f(x)$, we need to determine which part of the function to use for each value of $x$.

The function is defined as:

• $f(x) = 4$ if $x \leq 3$
• $f(x) = 3x - 5$ if $x > 3$

Now, let's evaluate $f(-4)$, $f(0)$, $f(3)$, $f(4)$, and $f(6)$:

1. For $f(-4)$:

   • Since $-4 \leq 3$, we use $f(x) = 4$.
   • $f(-4) = 4$

2. For $f(0)$:

   • Since $0 \leq 3$, we use $f(x) = 4$.
   • $f(0) = 4$

3. For $f(3)$:

   • Since $3 \leq 3$, we use $f(x) = 4$.
   • $f(3) = 4$

4. For $f(4)$:

   • Since $4 > 3$, we use $f(x) = 3x - 5$.
   • $f(4) = 3(4) - 5 = 12 - 5 = 7$

5. For $f(6)$:

   • Since $6 > 3$, we use $f(x) = 3x - 5$.
   • $f(6) = 3(6) - 5 = 18 - 5 = 13$

Summary of Results:

• $f(-4) = 4$
• $f(0) = 4$
• $f(3) = 4$
• $f(4) = 7$
• $f(6) = 13$

If you have any questions or need further clarification, feel free to ask!

Here are some related questions to consider:

1. What happens to $f(x)$ as $x$ approaches 3 from the left and the right?
2. How would the function change if we altered the conditions?
3. Can you plot the function to visualize its behavior?
4. What is the range of $f(x)$ based on this piecewise definition?
5. How do piecewise functions apply in real-world situations?

Tip: Always carefully check the intervals in piecewise functions to determine which formula to use for evaluation.