Let's analyze the statements based on the graph provided:

1. Statement (a): The domain is $(-∞, 4]$

   • The graph shown appears to be a parabola, opening downwards, which suggests a domain that covers all $x$-values from $-∞$ to $+∞$. Hence, statement (a) is incorrect.

2. Statement (b): The range is $(-∞, 4]$

   • The maximum $y$-value of the graph is 4, and it extends downward to $-∞$. This makes the range $(-∞, 4]$. Thus, statement (b) is correct.

3. Statement (c): The graph is positive on the interval $(0, ∞)$

   • The graph is positive (above the $x$-axis) only from $x = -2$ to $x = 2$. Since this does not match the interval $(0, ∞)$, statement (c) is incorrect.

4. Statement (d): The graph is decreasing on the interval $(-1, ∞)$

   • The graph begins decreasing from the vertex at $x = 0$ to $\infty$. However, it is not decreasing on $(-1, ∞)$ because it increases from $-∞$ to $x = 0$. Therefore, statement (d) is incorrect.

Summary:

Only statement (b) is correct.

Would you like a more detailed breakdown of each step or have any specific questions?

Here are some follow-up questions to deepen your understanding:

1. What does it mean for a function to be positive on an interval?
2. How can we determine the range of a function by looking at its graph?
3. Why is the domain of a parabola generally all real numbers?
4. How can we identify intervals where a graph is increasing or decreasing?
5. What would change in the answer if the parabola opened upwards instead of downwards?

Tip: When analyzing the domain and range of a function from its graph, remember the domain covers all possible $x$-values, while the range covers all possible $y$-values.