https://file.aiquickdraw.com/mathbot/file/1726073835495ecinsauh.jpg

Let's analyze the graph in the image and address each question.

### (a) **Domain of f**:
The domain is the set of all possible input values (x-values) for the function. Based on the graph, it appears that the function extends indefinitely in both directions along the x-axis. Hence, the domain of \( f \) is:
\[
\text{Domain: } (-\infty, \infty)
\]

### (b) **Range of f**:
The range is the set of all output values (y-values) the function can take. From the graph, the lowest point is at \( y = -2 \), and the function extends upwards without bound. Thus, the range of \( f \) is:
\[
\text{Range: } [-2, \infty)
\]

### (c) **X-intercepts**:
X-intercepts are the points where the graph crosses the x-axis. From the graph, the x-intercepts are at \( x = -3 \) and \( x = 1 \). So the x-intercepts are:
\[
x = -3, x = 1
\]

### (d) **Y-intercept**:
The y-intercept is the point where the graph crosses the y-axis. This occurs at \( y = -1 \). Therefore, the y-intercept is:
\[
y = -1
\]

### (e) **Intervals on which f is increasing**:
The function is increasing when the graph is going upwards as we move from left to right. From the graph, the function increases on the interval \( (x = -1, \infty) \). So, the interval is:
\[
(-1, \infty)
\]

### (f) **Intervals on which f is decreasing**:
The function is decreasing when the graph is going downwards as we move from left to right. From the graph, the function decreases on the interval \( (-\infty, -1) \). So, the interval is:
\[
(-\infty, -1)
\]

### (g) **Intervals on which f is constant**:
There are no intervals where the function is constant (i.e., horizontal). Hence, there are no such intervals.

### (h) **The number at which f has a relative minimum**:
A relative minimum occurs at the lowest point in a particular section of the graph. The graph has a relative minimum at \( x = -1 \).

### (i) **The relative minimum of f**:
The value of the relative minimum is the y-coordinate at \( x = -1 \), which is \( y = -2 \). So, the relative minimum value is:
\[
y = -2
\]

### (j) **f(-4)**:
To find \( f(-4) \), we check the y-value of the function when \( x = -4 \). From the graph, \( f(-4) = 0 \).

### (k) **The values of x for which \( f(x) = 3 \)**:
To solve \( f(x) = 3 \), we look for the points on the graph where the y-value is 3. There are no points on the graph where \( f(x) = 3 \), so there are no such values of \( x \).

### (l) **Is f even, odd, or neither?**:
A function is **even** if it is symmetric with respect to the y-axis, and **odd** if it is symmetric with respect to the origin. The given function is not symmetric with respect to the y-axis or the origin, so it is **neither even nor odd**.

---

Would you like further details or have any questions on this? Here are 5 related questions that could deepen your understanding:

1. How can you tell whether a function is increasing or decreasing just from its graph?
2. Why is the range of the function restricted to values greater than or equal to -2?
3. What is the significance of x- and y-intercepts in the context of a function?
4. How do relative minimum and maximum points help in analyzing functions?
5. How can you determine whether a function is even, odd, or neither from its equation?

**Tip**: When solving for intercepts, remember that for x-intercepts, set \( y = 0 \), and for y-intercepts, set \( x = 0 \).

Use the graph to find: (a) the domain of f, (b) the range of f, (c) the x-intercepts, (d) the y-intercept, (e) intervals on which f is increasing, (f) intervals on which f is decreasing, (g) intervals on which f is constant, (h) the number at which f has a relative minimum, (i) the relative minimum of f, (j) f(-4), (k) the values of x for which f(x) = 3, (l) Is f even, odd, or neither?

Learn how to analyze a function from a graph by finding its domain, range, intercepts, increasing and decreasing intervals, and more. This problem is suited for students in Grades 9-11, covering key function analysis concepts.

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