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Use the graph of f to determine each of the following.
​(a) the domain of f
​(b) the range of f
​(c) the zeros of f
​(d) f left parenthesis 0 right parenthesisf(0)
​(e) the intervals on which f is increasing
​(f) the intervals on which f is decreasing
​(g) the values for which ​f(x)less than or equals≤0
​(h) any relative maxima or minima
​(i) the​ value(s) of x for which f left parenthesis x right parenthesis equals 4f(x)=4
​(j) Is f left parenthesis negative 2 right parenthesisf(−2) positive or​ negative?

Let's analyze the graph provided step by step to answer each of the questions about the function \( f(x) \).

### (a) Domain of \( f \)
The domain is the set of all possible input values (x-values) for which the function is defined. From the graph, the function starts at \( x = -8 \) and continues to \( x = 4 \). Therefore, the domain is:
\[
\text{Domain: } [-8, 4]
\]

### (b) Range of \( f \)
The range is the set of all possible output values (y-values). From the graph, the function has a maximum value of \( y = 4 \) and goes down to \( y = -5 \). Therefore, the range is:
\[
\text{Range: } [-5, 4]
\]

### (c) Zeros of \( f \)
Zeros are the x-values where \( f(x) = 0 \) (i.e., where the graph crosses the x-axis). From the graph, the zeros occur at \( x = -6 \) and \( x = 2 \).
\[
\text{Zeros: } x = -6, 2
\]

### (d) \( f(0) \)
To find \( f(0) \), locate where \( x = 0 \) on the graph and find the corresponding \( y \)-value. From the graph:
\[
f(0) = 2
\]

### (e) Intervals where \( f \) is increasing
The function is increasing where the graph is going upward. This occurs from \( x = -8 \) to \( x = -4 \) (since the graph rises from left to right in this interval).
\[
\text{Increasing on: } [-8, -4]
\]

### (f) Intervals where \( f \) is decreasing
The function is decreasing where the graph is going downward. This occurs from \( x = -4 \) to \( x = 4 \).
\[
\text{Decreasing on: } [-4, 4]
\]

### (g) Values where \( f(x) \leq 0 \)
The values of \( x \) where the graph is below or on the x-axis (i.e., \( f(x) \leq 0 \)) are from \( x = -8 \) to \( x = -6 \), and from \( x = 2 \) to \( x = 4 \).
\[
f(x) \leq 0 \text{ for } x \in [-8, -6] \cup [2, 4]
\]

### (h) Relative maxima or minima
- **Relative maximum:** The highest point within a certain interval. The relative maximum occurs at \( x = -4 \), where \( f(x) = 4 \).
- **Relative minimum:** There is no relative minimum, as the function does not have any lower turning points before continuing downward.

### (i) Values of \( x \) for which \( f(x) = 4 \)
The value of \( f(x) = 4 \) occurs at \( x = -4 \).
\[
f(x) = 4 \text{ at } x = -4
\]

### (j) Is \( f(-2) \) positive or negative?
To find \( f(-2) \), locate \( x = -2 \) on the graph and find the corresponding \( y \)-value. From the graph, \( f(-2) \) is positive because the graph is above the x-axis at \( x = -2 \).
\[
f(-2) \text{ is positive.}
\]

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Would you like further details or explanations? Here are some related questions to consider:

1. How do you determine the relative maxima and minima of a function from its graph?
2. What is the difference between absolute maxima/minima and relative maxima/minima?
3. How can the intervals of increase and decrease be found using the derivative of a function?
4. How would you describe the behavior of the function at \( x = 4 \) where the function terminates?
5. Can a function have more than one relative maximum?

**Tip:** A function's domain refers to the x-values for which the function is defined, and the range refers to the y-values the function can take. Understanding this distinction is key when interpreting graphs.

Use the graph of f to determine each of the following.
(a) the domain of f
(b) the range of f
(c) the zeros of f
(d) f(0)
(e) the intervals on which f is increasing
(f) the intervals on which f is decreasing
(g) the values for which f(x) ≤ 0
(h) any relative maxima or minima
(i) the values of x for which f(x) = 4
(j) Is f(-2) positive or negative?

This problem involves analyzing the graph of a quadratic function to determine its domain, range, zeros, intervals of increase/decrease, and more. Suitable for high school students, this solution explores key concepts of function behavior from graphical data.

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