The question asks about the range and domain of the given piecewise function, $f(x)$, as shown in the graph. Let's analyze the options based on the graph:

Step 1: Domain of the function

The domain refers to all the possible $x$-values for which the function is defined. From the graph:

• The function starts at $x = 0$ and continues to $x = 8$, without any breaks in between. Thus, the domain of $f(x)$ is $0 \leq x \leq 8$.

Step 2: Range of the function

The range refers to all the possible $y$-values that $f(x)$ can take. From the graph:

• The function appears to have three distinct output values: 2, 3, and 4. Thus, the range of $f(x)$ is $2 \leq y \leq 4$.

Step 3: Analyzing the Answer Choices:

1. The range of $f(x)$ is $2 \leq y \leq 4$.

   • This is correct based on the graph.

2. The domain of $f(x)$ is $0 \leq x \leq 8$.

   • This is correct as well.

3. The range of $f(x)$ is all positive real integers from 2 to 4.

   • This is incorrect because the range includes only $2, 3,$ and $4$, which are integers, not all positive real numbers.

4. The domain of $f(x)$ is all positive real integers between 1 and 8.

   • This is incorrect because the domain includes all real numbers between $0 \leq x \leq 8$, not just integers.

Correct Statements:

• The first two options are correct.

Let me know if you want further details, and here are some follow-up questions for deeper understanding:

1. How can you determine the domain of a function from its graph?
2. What is the difference between a function's range and its codomain?
3. How would the domain change if the graph included an open circle at $x = 8$?
4. What are piecewise functions, and why are they important in modeling real-world scenarios?
5. How can a function's range be restricted or extended by its mathematical expression?

Tip: Always pay attention to whether a graph includes solid or open circles, as they can indicate whether an endpoint is included in the domain or range.