The problem has two parts. The first part deals with a linear function $f(x)$ represented by the line $(D)$. We are given a table with some $x$ values and corresponding $f(x)$ values, with some missing entries that we need to fill in. Then, we need to determine the expression for $f(x)$. The second part deals with another linear function $g(x)$ represented by the line $(\Delta)$. We need to find the coefficient of $g(x)$ and then determine the expression for $g(x)$.

1. Problem Description

The problem has two parts. The first part deals with a linear function $f(x)$ represented by the line $(D)$. We are given a table with some $x$ values and corresponding $f(x)$ values, with some missing entries that we need to fill in. Then, we need to determine the expression for $f(x)$. The second part deals with another linear function $g(x)$ represented by the line $(\Delta)$. We need to find the coefficient of $g(x)$ and then determine the expression for $g(x)$.

2. Solution Steps

Part 1:

a) Completing the table for $f(x)$.

We are given that $f(2) = 3$ and $f(-3) = -3$. Since $f(x)$ is a linear function, we can write it as $f(x) = ax$ for some constant $a$.

Using the given information, $f(2) = 2a = 3$, so $a = \frac{3}{2}$. Thus, $f(x) = \frac{3}{2}x$.

Now we can find the missing values in the table:

If $x = -2$, then $f(-2) = \frac{3}{2}(-2) = -3$.

We already have $f(2)=3$ and $f(-3) = -3$

The completed table is:

| $x$ | 2 | -2 | -3 |

| ------- | ---- | -- | ---- |

| $f(x)$ | 3 | -3 | -3 |

b) Determining the expression for $f(x)$.

As derived above, $f(x) = \frac{3}{2}x$.

Part 2:

Finding the coefficient of $g(x)$ and the expression for $g(x)$.

Since the line $(\Delta)$ passes through the origin, the function $g(x)$ is of the form $g(x) = bx$ for some constant $b$.

From the graph, we can see that the line $(\Delta)$ passes through the point $(1.5, -3)$. So, we have $g(1.5) = -3$. Thus, $1.5b = -3$, which means $b = \frac{-3}{1.5} = -2$.

Therefore, $g(x) = -2x$.

The coefficient of $g(x)$ is $-2$.

3. Final Answer

Part 1:

a) The completed table is:

| $x$ | 2 | -2 | -3 |

| ------- | ---- | -- | ---- |

| $f(x)$ | 3 | -3 | -3 |

b) The expression for $f(x)$ is $f(x) = \frac{3}{2}x$.

Part 2:

The coefficient of $g(x)$ is $-2$. The expression for $g(x)$ is $g(x) = -2x$.