Let's break down problem 32 from the image, which asks about the following for the graph:

(a) The domain and range of the function $f$:

• The domain is the set of all possible x-values for which the function is defined.
• The range is the set of all possible y-values that the function attains.

(b) The intervals on which $f$ is increasing and on which $f$ is decreasing:

• A function is increasing on intervals where, as $x$ increases, $f(x)$ also increases.
• A function is decreasing on intervals where, as $x$ increases, $f(x)$ decreases.

(a) Domain and Range:

From the graph in problem 32:

• The domain of $f$ appears to be all $x$-values from $-1$ to $2$ because the graph starts at $x = -1$ and ends at $x = 2$. Therefore, the domain is: $\text{Domain: } [-1, 2]$
• The range of $f$ (the $y$-values it attains) appears to go from a minimum value of $-1$ to a maximum value of $1$. So the range is: $\text{Range: } [-1, 1]$

(b) Intervals of Increasing and Decreasing:

Looking at the shape of the graph:

• The function $f(x)$ is increasing from $x = -1$ to $x = 1$, since the graph goes upward in this interval.
• The function $f(x)$ is decreasing from $x = 1$ to $x = 2$, as the graph goes downward after this point.

Thus:

• $f(x)$ is increasing on the interval $[-1, 1]$.
• $f(x)$ is decreasing on the interval $[1, 2]$.

Would you like to explore this further or see more details about intervals? Here are five questions to consider:

1. How can we confirm the turning points of a graph using derivatives?
2. How does the concept of continuity relate to the domain of a function?
3. How do we find the exact range of a function from its graph or equation?
4. What real-world situations can be modeled by increasing and decreasing functions?
5. Can a function have multiple intervals of increase and decrease? How would that look graphically?

Tip: When analyzing increasing and decreasing intervals, always look for the points where the slope changes from positive to negative or vice versa!