https://file.aiquickdraw.com/mathbot/file/1725673547404l5ysf859.jpg

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:

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### 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 \( x = 5 \)

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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.

Given the graph of the function g(x), answer the following questions:
(a) Find the domain and range of the function g(x).
(b) Find the rule of correspondence for g(x) when g(x) = 1.
(c) Find the rule of correspondence for g(x) = -4.
(d) Find the set of all values of x such that g(x) = 0.
(e) Find the set of all values for which the function g(x) is increasing.
(f) Is the function g(x) injective? Justify your answer.
(g) Find the absolute maximum and minimum of g(x).
(h) Determine the function's behavior as x -> ∞ and x -> -∞.
(i) Discuss whether the function has any vertical or horizontal asymptotes.

Explore the graph of the function g(x) to determine its domain, range, zeros, and intervals of increase. This problem also covers injectivity, maximum and minimum values, and asymptotic behavior. Suitable for high school students in Grades 10-12.

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