The problem asks us to evaluate the function $f(x) = 3 \left\lfloor x - 3 \right\rfloor + 2$ at two different values of $x$: 1.6 and -3.3. Here, the symbol $\left\lfloor x \right\rfloor$ represents the floor function, which gives the greatest integer less than or equal to $x$.

Step-by-Step Solution:

1. Evaluate $f(1.6)$:

   • First, compute $x - 3$ for $x = 1.6$:
     $1.6 - 3 = -1.4$
   • Now, apply the floor function $\left\lfloor -1.4 \right\rfloor$, which gives $-2$, because $-2$ is the greatest integer less than or equal to $-1.4$.
   • Substitute this into the function:
     $f(1.6) = 3(-2) + 2 = -6 + 2 = -4$.

2. Evaluate $f(-3.3)$:

   • First, compute $x - 3$ for $x = -3.3$:
     $-3.3 - 3 = -6.3$
   • Now, apply the floor function $\left\lfloor -6.3 \right\rfloor$, which gives $-7$, because $-7$ is the greatest integer less than or equal to $-6.3$.
   • Substitute this into the function:
     $f(-3.3) = 3(-7) + 2 = -21 + 2 = -19$.

Final Answers:

• $f(1.6) = -4$
• $f(-3.3) = -19$

Would you like further explanations or examples?

Here are some follow-up questions to explore more about floor functions and this problem:

1. What is the difference between the floor function and the ceiling function?
2. How would the function change if we replaced the floor function with the ceiling function?
3. Can you calculate the value of $f(3.5)$?
4. What happens to the value of $f(x)$ as $x$ increases past 3?
5. How would you solve $f(x) = 0$ for $x$?

Tip: The floor function $\left\lfloor x \right\rfloor$ can sometimes be tricky. To make sure you get the correct answer, always find the nearest integer less than or equal to $x$.