To determine the intervals on which $f(x) \geq 0$ from the given graph, we need to identify the sections of the $x$-axis where the function $f(x)$ is either positive (above the $x$-axis) or zero (on the $x$-axis).

Let's analyze the graph:

1. Identify where $f(x) = 0$: These are the points where the graph crosses the $x$-axis.
2. Locate intervals above the $x$-axis: These are the intervals where the graph is positive, meaning $f(x) > 0$.

Observing the Graph

From the graph, it appears that:

• $f(x) = 0$ at approximately $x = -5$, $x = -3$, $x = 1$, $x = 4$, and $x = 7$.
• The graph is above the $x$-axis (i.e., $f(x) \geq 0$) on the following intervals:
  • $x \in [-5, -3]$
  • $x \in [1, 4]$
  • $x \in [7, \infty)$

Final Answer

The intervals where $f(x) \geq 0$ are: [ [-5, -3] \cup [1, 4] \cup [7, \infty) ]

Would you like further details on this solution or have any questions?

Related Questions

1. How would the answer change if the question asked for $f(x) > 0$ instead of $f(x) \geq 0$?
2. How do we determine where a function is decreasing or increasing on a graph?
3. What is the difference between intervals where $f(x) \geq 0$ and where $f(x) > 0$?
4. How would we find the intervals where $f(x) \leq 0$ for this graph?
5. Can a polynomial function like this one have an infinite number of roots?

Tip

Always check the points where the function touches or crosses the $x$-axis to determine intervals of non-negativity or non-positivity.