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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.

Use the graphing tool to determine the true statements regarding the function represented in the graph. Check all that apply. The options are:
1. f(x) > 0 over the interval (1, ∞)
2. f(x) ≤ 0 over the interval [1, ∞)
3. f(x) ≤ 0 over the interval (-∞, 1]
4. f(x) > 0 over the interval (-∞, 1)
5. f(x) > 0 over the interval (-∞, ∞)

Analyze the behavior of the function based on its graph, focusing on identifying intervals where the function is positive, negative, or zero. This example covers inequalities, graph interpretation, and function behavior for students in Grades 9-12.

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