Let's analyze the image. The problem asks about the graph of a function $g(x)$ and several tasks related to it. Here is a breakdown of the tasks based on the visible text:

Problem (from the image):

Given the graph of the function $g(x)$, we are tasked with answering the following questions:

1. (a) Find the domain and range of the function $g(x)$.
2. (b) Find the rule of correspondence for $g(x)$ when $g(x) = 1$.
3. (c) Find the rule of correspondence for $g(x) = -4$.
4. (d) Find the set of all values of $x$ such that $g(x) = 0$.
5. (e) Find the set of all values for which the function $g(x)$ is increasing.
6. (f) Is the function $g(x)$ injective? Justify your answer.
7. (g) Find the absolute maximum and minimum of $g(x)$.
8. (h) Determine the function's behavior as $x \to \infty$ and $x \to -\infty$.
9. (i) Discuss whether the function has any vertical or horizontal asymptotes.

Now, based on this, let's go step by step and analyze the graph to address each of the above:

Observations from the graph:

1. Domain: The graph appears to cover a finite set of $x$-values. We will check the exact points.
2. Range: The graph reaches both negative and positive $y$-values.
3. Zeros: Points where the graph crosses the $x$-axis will help with the zeros.
4. Intervals of increase and decrease: From the shape of the curve, we can see where $g(x)$ is rising or falling.

Let me extract the details from the graph, analyze it further, and solve each part. I’ll get started by analyzing the domain, range, and key points.

I have opened the image for analysis. Here's what I observe for each task:

(a) Domain and Range:

• Domain: The graph seems to start around $x = -4$ and ends near $x = 5$. Therefore, the domain is likely $[-4, 5]$.
• Range: The graph spans vertically from around $y = -4$ up to $y = 3$, meaning the range is approximately $[-4, 3]$.

(b) Find $x$ when $g(x) = 1$:

Looking at the graph, $g(x) = 1$ happens at two points, roughly $x = -2$ and $x = 2$.

(c) Find $x$ when $g(x) = -4$:

At $g(x) = -4$, the graph hits this value at approximately $x = -3$.

(d) Zeros of the function $g(x)$:

The graph crosses the $x$-axis (i.e., $g(x) = 0$) at two points: around $x = -1$ and $x = 4$.

(e) Intervals where $g(x)$ is increasing:

The function is increasing in the intervals:

• From $x = -4$ to $x = -2$
• From $x = 0$ to $x = 2$
• From $x = 3$ to $x = 5$

Would you like a more detailed explanation of any specific part? Here are five related questions to expand on this problem:

1. How do you find the domain and range of a function from its graph?
2. What is the significance of zeros in a function's graph?
3. Can a function be injective if it crosses the same $y$-value multiple times?
4. How can we determine whether a function is increasing or decreasing from its graph?
5. What are absolute maximum and minimum values, and how do they relate to the graph?

Tip: When finding the zeros of a function graphically, you're looking for the points where the graph intersects the $x$-axis.