Let's break down each part of the question based on the graph description you provided:

(a) Increasing Interval(s):

The function is increasing when the graph rises as you move from left to right. From the description:

• There is a curve that rises from right to left between the points (-2, 1) and (-4, 5), which indicates an increasing part of the graph.

• Since the graph increases between these two points, the open interval where the function is increasing is:

  Interval: $(-4, -2)$

Answer for part (a): The function is increasing on the open interval $(-4, -2)$.

---

(b) Decreasing Interval(s):

The function is decreasing when the graph falls as you move from left to right. Based on the description:

• The third branch of the graph falls from left to right between the points (4, 0.5) and (6, -3.5), which means the function is decreasing between these points.

Interval: $(4, 6)$

Answer for part (b): The function is decreasing on the open interval $(4, 6)$.

---

(c) Constant Interval(s):

The function is constant when the graph is a horizontal line (no change in y-value). According to the description:

• The second branch is a horizontal line segment from the open circle at $(-2, 2)$ to the open circle at $(4, 2)$. Since this is a flat, horizontal line, the function is constant on this interval.

Interval: $(-2, 4)$

Answer for part (c): The function is constant on the open interval $(-2, 4)$.

---

(d) Domain of the Function:

The domain consists of all the x-values for which the function is defined. According to the graph description:

• The function is defined for all $x$-values from $-4$ to $6$, excluding the points where open circles are placed at $(-2, 2)$ and $(4, 2)$, indicating that these points are not part of the domain.

Domain: $(-4, 6)$

Answer for part (d): The domain of the function is $(-4, 6)$.

---

(e) Range of the Function:

The range consists of all the possible y-values that the function can take. Based on the description:

• The y-values range from $-3.5$ to $5$ (inclusive of the lower and upper bounds for the curves and horizontal segment).

Range: $[-3.5, 5]$

Answer for part (e): The range of the function is $[-3.5, 5]$.

---

Summary of Answers:

1. (a) The function is increasing on the open interval $(-4, -2)$.
2. (b) The function is decreasing on the open interval $(4, 6)$.
3. (c) The function is constant on the open interval $(-2, 4)$.
4. (d) The domain of the function is $(-4, 6)$.
5. (e) The range of the function is $[-3.5, 5]$.

Would you like to dive deeper into any specific part, or do you have any further questions? Here are some related questions for further practice:

1. How can you determine where a function is increasing or decreasing just by looking at the graph?
2. Can a function be constant for more than one interval?
3. How do open circles on a graph affect the domain and range?
4. Why is the domain not defined at $x = -2$ and $x = 4$?
5. How do you find the range of a function from its graph?

Tip: When analyzing a graph for intervals of increase, decrease, or constancy, look for sections where the graph either slopes upwards (increasing), slopes downwards (decreasing), or remains flat (constant).