https://file.aiquickdraw.com/mathbot/file/1725652835730nha2gp7v.jpg

Consider the function 𝑦=𝑓(𝑥) given in the following graph.

Click on the graph to view the enlarged graph

Use the graph of 𝑓 to determine the following.

The domain of 𝑓: functionsequation editor

The range of 𝑓: functionsequation editor

Note: Use INF or -INF for ∞ or −∞, respectively.
Make sure your answer is in interval notation.
Enter "{ }" without quotes if there is no solution (the empty set).
Use the letter U for union. e.g., an answer of 𝑥<2 or 𝑥≥6
should be entered as (-inf,2) U [6,inf), etc.

The zeros of 𝑓: {functionsequation editor }

𝑓(0)= functionsequation editor

The interval(s) on which 𝑓 is increasing: functionsequation editor

The interval(s) on which 𝑓 is decreasing: functionsequation editor

Note: Use INF or -INF for ∞ or −∞, respectively.
Make sure your answer is in interval notation.
Enter "NONE" without quotes if there are no intervals where the function is decreasing, increasing, or constant.
Separate multiple intervals with a comma. e.g., if the function is decreasing on both (2,3) and (4,5), then your answer would be (2,3),(4,5).

The values of 𝑥 for which 𝑓(𝑥)≥0: functionsequation editor

The number at which 𝑓 has a relative minimum: functionsequation editor

The relative minimum of 𝑓: functionsequation editor

The value of 𝑥 for which 𝑓(𝑥)=−9 is: functionsequation editor

Is 𝑓(−3) positive or negative?

A. Positive
B. Negative

Let's analyze the graph to answer the questions step-by-step.

### 1. **Domain of \( f \):**
   The domain is the set of all possible \( x \)-values for which the function is defined. Looking at the graph, we see that the function extends infinitely in both directions for \( x \). Therefore, the domain is:
   \[
   (-\infty, \infty)
   \]

### 2. **Range of \( f \):**
   The range is the set of all possible \( y \)-values. The function seems to go infinitely upwards as \( x \) decreases and has a minimum point at \( y = -9 \). Therefore, the range is:
   \[
   [-9, \infty)
   \]

### 3. **Zeros of \( f \):**
   The zeros of the function are the points where the graph crosses the \( x \)-axis. From the graph, this happens at \( x = -2 \) and \( x = 4 \). Thus, the zeros are:
   \[
   \{-2, 4\}
   \]

### 4. **\( f(0) \):**
   To find \( f(0) \), look at the value of the function when \( x = 0 \). From the graph, \( f(0) = 2 \).

### 5. **Intervals where \( f \) is Increasing:**
   The function is increasing when its graph moves upwards as we move from left to right. From the graph, \( f(x) \) is increasing on two intervals:
   \[
   (-\infty, -2), \; (2, 6)
   \]

### 6. **Intervals where \( f \) is Decreasing:**
   The function is decreasing when its graph moves downwards as we move from left to right. From the graph, \( f(x) \) is decreasing on the interval:
   \[
   (-2, 2)
   \]

### 7. **Values of \( x \) for which \( f(x) \geq 0 \):**
   \( f(x) \) is non-negative (i.e., \( f(x) \geq 0 \)) where the graph is on or above the \( x \)-axis. This occurs in two intervals:
   \[
   (-\infty, -2] \cup [4, 6]
   \]

### 8. **Number at which \( f \) has a Relative Minimum:**
   The relative minimum is the lowest point on the graph within a certain interval. The graph shows a relative minimum at \( x = 2 \).

### 9. **Relative Minimum of \( f \):**
   The relative minimum value of \( f(x) \) at \( x = 2 \) is \( -9 \).

### 10. **Value of \( x \) for which \( f(x) = -9 \):**
   The function reaches \( -9 \) at \( x = 2 \). So, the value of \( x \) is:
   \[
   x = 2
   \]

### 11. **Is \( f(-3) \) Positive or Negative?**
   At \( x = -3 \), the graph is above the \( x \)-axis, which means \( f(-3) \) is positive.

### **Answer: A. Positive**

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If you have any further questions or need more details, let me know!

### Related Questions:
1. How do we determine the concavity of a function from its graph?
2. What is the difference between absolute and relative extrema?
3. How do you find the intervals of concavity and points of inflection?
4. How can you determine the end behavior of a function from its graph?
5. What does it mean for a function to be continuous on an interval?

### Tip:
When analyzing graphs of functions, always look for key features such as intercepts, extrema, and asymptotic behavior to fully understand the function's properties.

Consider the function y=f(x) given in the graph. Use the graph of f to determine the following:
- The domain of f.
- The range of f.
- The zeros of f.
- f(0).
- The interval(s) on which f is increasing and decreasing.
- The values of x for which f(x)≥0.
- The relative minimum of f.
- The value of x for which f(x)=-9.
- Is f(-3) positive or negative?

Analyze the function y=f(x) from its graph to find the domain, range, zeros, f(0), increasing and decreasing intervals, and the relative minimum. Suitable for high school students (Grades 9-12).

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