To solve the problem given in the image, let’s go through each part systematically.

1. Problem Statement Overview:
   • Two functions are defined:
     • $f(x) = 3x + 3$
     • $g(x) = \frac{x - 3}{3}$
   • We are tasked with solving several questions based on these functions.

Step-by-Step Solutions

(ক) $g^{-1}(-3)$ এর মান নির্ণয় কর

To find $g^{-1}(-3)$, we need to determine the inverse of $g(x)$, then plug in $-3$ to find the value.

1. Find $g^{-1}(x)$:

   • Start with $y = g(x) = \frac{x - 3}{3}$.
   • Rearrange to solve for $x$ in terms of $y$: $y = \frac{x - 3}{3} \Rightarrow x - 3 = 3y \Rightarrow x = 3y + 3$
   • Thus, the inverse function is $g^{-1}(x) = 3x + 3$.

2. Calculate $g^{-1}(-3)$:

   • Substitute $-3$ into $g^{-1}(x)$: $g^{-1}(-3) = 3(-3) + 3 = -9 + 3 = -6$
   • Therefore, $g^{-1}(-3) = -6$.

(খ) $f(x)$ সার্বিক ক্রমহ্রাসশীল বা ক্রমবর্ধমান কিনা তা নির্ণয় কর

To determine if $f(x) = 3x + 3$ is increasing or decreasing, we examine its derivative:

1. Find $f'(x)$: $f(x) = 3x + 3 \Rightarrow f'(x) = 3$
2. Since $f'(x) = 3 > 0$, $f(x)$ is an increasing function on $\mathbb{R}$.

(গ) দেখাও যে, $g = f^{-1}$

To show that $g$ is the inverse of $f$, we need to confirm that $f(g(x)) = x$ and $g(f(x)) = x$:

1. Calculate $f(g(x))$:
   • Substitute $g(x) = \frac{x - 3}{3}$ into $f$: $f(g(x)) = f\left(\frac{x - 3}{3}\right) = 3 \cdot \frac{x - 3}{3} + 3 = (x - 3) + 3 = x$
2. Calculate $g(f(x))$:
   • Substitute $f(x) = 3x + 3$ into $g$: $g(f(x)) = g(3x + 3) = \frac{3x + 3 - 3}{3} = \frac{3x}{3} = x$
3. Since both $f(g(x)) = x$ and $g(f(x)) = x$, we conclude that $g = f^{-1}$.

Final Answers

1. (ক) $g^{-1}(-3) = -6$
2. (খ) $f(x)$ is an increasing function.
3. (গ) It is shown that $g = f^{-1}$.

Would you like further details on any part of the solution?

Follow-Up Questions:

1. How do we determine if a function is increasing or decreasing using derivatives?
2. What are the steps to find the inverse of a function?
3. How can we verify that two functions are inverses of each other?
4. Why is the derivative of a linear function like $f(x) = 3x + 3$ constant?
5. What are some practical applications of finding inverse functions?

Tip: To verify if two functions are inverses, always check that applying one function after the other returns the original input value.