https://file.aiquickdraw.com/mathbot/file/1725384805652f9pyjlsz.jpg

The image shows a problem related to a piecewise linear function, where the graph of \( y = f(x) \) is depicted. Below the graph, several questions are listed, which ask for various properties of the function \( f(x) \).

Here’s a summary of the questions based on the content visible in the image:

1. **(a)** Find \( f(0) \).
2. **(b)** Find \( f(3) \).
3. **(c)** Find \( f(4) \).
4. **(d)** Find the values of \( x \) for which \( f(x) = 3 \).
5. **(e)** Find the domain of \( f(x) \).
6. **(f)** Find the range of \( f(x) \).
7. **(g)** Find the intervals on which the graph of \( f(x) \) is increasing.
8. **(h)** Find the intervals on which the graph of \( f(x) \) is decreasing.
9. **(i)** State the equation of the function \( f(x) \).

Let's go through these one by one, using the information from the graph:

### (a) Find \( f(0) \)
- From the graph, locate the point where \( x = 0 \). The corresponding \( y \)-value will be \( f(0) \).

### (b) Find \( f(3) \)
- Locate the point where \( x = 3 \) on the graph. The corresponding \( y \)-value is \( f(3) \).

### (c) Find \( f(4) \)
- Similarly, find the point where \( x = 4 \) on the graph, and determine the \( y \)-value, which is \( f(4) \).

### (d) Find the values of \( x \) for which \( f(x) = 3 \)
- Look for the points on the graph where the \( y \)-value equals 3, then note the corresponding \( x \)-values.

### (e) Find the domain of \( f(x) \)
- The domain is the set of all \( x \)-values for which \( f(x) \) is defined. Check the range of \( x \)-values covered by the graph.

### (f) Find the range of \( f(x) \)
- The range is the set of all \( y \)-values that \( f(x) \) takes. Examine the graph to determine the lowest and highest \( y \)-values.

### (g) Find the intervals on which the graph of \( f(x) \) is increasing
- The function is increasing where the graph has a positive slope. Identify these intervals on the graph.

### (h) Find the intervals on which the graph of \( f(x) \) is decreasing
- Similarly, find where the graph has a negative slope to determine where the function is decreasing.

### (i) State the equation of the function \( f(x) \)
- Based on the graph, determine the piecewise equations for different intervals of \( x \).

If you would like, I can go through each part with more specific calculations based on the visible parts of the graph, or answer any specific question you may have. 

Do you want to explore these details further? If so, which part would you like to start with?

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**Here are five questions related to this topic:**

1. How can you determine if a piecewise function is continuous at a certain point?
2. What is the difference between the domain and range of a function?
3. How do you find the slope of a line segment in a piecewise function?
4. Can a function be increasing on one interval and decreasing on another? Explain with an example.
5. How do you determine the equation of a linear segment in a piecewise function?

**Tip:** Always check the endpoints of the intervals when finding where a piecewise function is increasing or decreasing.

Find various properties of the piecewise function f(x) based on the given graph.

Learn how to analyze a piecewise function graph to find its domain, range, function values at specific points, and intervals of increase or decrease. This problem is suitable for students in Grades 10-12 and covers fundamental concepts in understanding piecewise functions.

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