https://file.aiquickdraw.com/mathbot/file/1727371832168hrxkjnvw.jpg

Let's go through the given questions step-by-step using the provided graph of \( f(x) \).

### (a) \( f(5) = \)
At \( x = 5 \), the value of \( f(x) \) is 2.

### (b) \( f(0) = \)
At \( x = 0 \), the value of \( f(x) \) is 4.

### (c) \( f(-4) = \)
At \( x = -4 \), the value of \( f(x) \) is 0.

### (d) \( f(-8) = \)
At \( x = -8 \), the value of \( f(x) \) is -6.

### (e) For what values of \( x \) is \( f(x) = 0 \)?
\( f(x) = 0 \) at \( x = -4, 0, \) and \( 8 \).

### (f) For what values of \( x \) is \( f(x) = 6 \)?
\( f(x) = 6 \) at \( x = 2 \).

### (g) What is the domain of \( f(x) \)?
The domain of \( f(x) \) is the set of all \( x \)-values for which \( f(x) \) is defined. From the graph, it is:
\[ [-8, 10] \]

### (h) What is the range of \( f(x) \)?
The range is the set of all \( y \)-values that \( f(x) \) takes. From the graph, the range is:
\[ [-6, 6] \]

### (i) Is \( f(x) \) a one-to-one function?
No, \( f(x) \) is not a one-to-one function because there are different \( x \)-values that correspond to the same \( y \)-value. For example, \( f(-4) = f(0) = 0 \).

### (j) In what interval is this function increasing?
The function is increasing in the intervals where as \( x \) increases, \( f(x) \) also increases:
\[ (-8, -2) \text{ and } (0, 2) \]

### (k) In what interval is this function decreasing?
The function is decreasing in the intervals where as \( x \) increases, \( f(x) \) decreases:
\[ (-2, 0), (2, 5), \text{ and } (5, 10) \]

### (l) Identify all local minimum points and values.
Local minimum points are where the function changes from decreasing to increasing:
1. \( (-2, -4) \)
2. \( (5, 2) \)

### (m) Identify all local maximum points and values.
Local maximum points are where the function changes from increasing to decreasing:
1. \( (0, 4) \)
2. \( (2, 6) \)

### (n) Find the global maximum if there is any.
The global maximum value is 6, which occurs at \( x = 2 \).

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If you need more details or have further questions, feel free to ask! Here are 5 follow-up questions based on this topic:

1. How can you determine if a function is one-to-one just by looking at the graph?
2. What is the difference between a local maximum and a global maximum?
3. How do you find the domain and range of more complex functions?
4. How can you use calculus to confirm the local maximum and minimum points?
5. What are the applications of knowing the increasing and decreasing intervals of a function?

**Tip:** For functions with multiple local extrema, drawing a sign chart of the derivative can help confirm where the function is increasing or decreasing.

Graph of the function f(x) is shown in the figure. Each square represents 1 unit by 1 unit. Determine the indicated values: f(5), f(0), f(-4), etc., as well as other questions about domain, range, extrema, and whether the function is one-to-one.

Analyze the graph of the function f(x) to find values for f(5), f(0), and f(-4), determine the domain and range, identify local maxima and minima, and assess increasing and decreasing intervals. Suitable for high school students learning function analysis.

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