The problem is asking about different intervals where the function $f(x)$ is positive, negative, or zero based on the given graph.

Let's analyze each statement:

1. $f(x) > 0$ over the interval $(1, \infty)$:

   • From the graph, we see that the function crosses the x-axis at $x = 1$ and becomes positive for $x > 1$. Therefore, this statement is true.

2. $f(x) \leq 0$ over the interval $[1, \infty)$:

   • This statement is false. The function is positive for values greater than $x = 1$, so it is not less than or equal to zero on this interval.

3. $f(x) \leq 0$ over the interval $(-\infty, 1]$:

   • This statement is true. The function is less than or equal to zero for all values from $-\infty$ up to $x = 1$, including $x = 1$.

4. $f(x) > 0$ over the interval $(-\infty, 1)$:

   • This is false because the function is negative for $x < 1$.

5. $f(x) > 0$ over the interval $(-\infty, \infty)$:

   • This statement is false. The function is not positive over the entire interval as it is negative for parts of $(-\infty, 1)$.

Correct choices:

• $f(x) > 0$ over the interval $(1, \infty)$.
• $f(x) \leq 0$ over the interval $(-\infty, 1]$.

Let me know if you'd like more details or have any questions! Here are five related questions you can explore next:

1. How can you determine the intervals of positivity and negativity from a graph?
2. What does it mean for a function to be continuous at a certain point?
3. How do you identify the critical points of a function from its graph?
4. What are the differences between strict inequalities ($>$, $<$) and non-strict inequalities ($\geq$, $\leq$) in graph analysis?
5. How can the behavior of a function at infinity be determined from a graph?

Tip: When analyzing graphs, always look for where the function crosses the x-axis to determine sign changes.