In the image, we have two graphs, and the question asks us to determine the interval(s) where $f(x) \geq 0$, meaning where the function is non-negative (greater than or equal to zero).

Let's analyze both graphs individually:

1. Left Graph:

   • The curve starts at some point on the x-axis (which looks around $x = -5$) and then goes down as $x$ increases.
   • Since the curve is above or touches the x-axis from $x = -\infty$ to around $x = -5$, the function is non-negative in this range.
   • In interval notation, this corresponds to $(-\infty, -5]$.

2. Right Graph:

   • This graph mirrors the left one but is positioned on the right side of the x-axis. The curve appears to start on the x-axis at around $x = 5$ and rises to the right.
   • Since the function is non-negative from $x = 5$ to $\infty$, the interval for the right graph is $[5, \infty)$.

Thus, the solution for the intervals where $f(x) \geq 0$ is: $(-\infty, -5] \cup [5, \infty)$

Let me know if you'd like further details or explanations!

5 Related Questions:

1. What is the significance of the intervals where $f(x) \geq 0$ in real-world applications?
2. How do you determine if a function is non-negative using its graph?
3. What other methods can you use to find where a function is non-negative besides using a graph?
4. How does interval notation change when dealing with inequalities involving $f(x) > 0$ instead of $f(x) \geq 0$?
5. How would the answer change if the curves touched the x-axis at different points?

Tip:

When analyzing graphs for intervals where $f(x) \geq 0$, always look for where the curve touches or stays above the x-axis!