Let's analyze the graph and the problem provided:

Step 1: Identifying $x$-intercepts (Solutions to $f(x) = 0$)

The $x$-intercepts are where the graph of $f(x)$ crosses the $x$-axis, meaning $f(x) = 0$ at those points.

From the graph:

• The curve crosses the $x$-axis at approximately $x = 1$, $x = 3$, and $x = 5$.

Thus, the solutions to $f(x) = 0$ are $x = 1, 3, 5$.

Step 2: Solving $f(x) \geq 0$

We need to find where the graph is on or above the $x$-axis (i.e., $f(x) \geq 0$).

Looking at the graph:

• The function is above or on the $x$-axis between $x = 1$ and $x = 3$, and again from $x = 5$ onward.

Thus, the solution to $f(x) \geq 0$ is:

• $[1, 3] \cup [5, \infty)$ in interval notation.

Answers:

• Solutions to $f(x) = 0$: $1, 3, 5$
• Solutions to $f(x) \geq 0$ in interval notation: $[1, 3] \cup [5, \infty)$

Would you like further clarification or details on any part of the solution?

Here are 5 related questions to deepen your understanding:

1. How would the solution change if the graph had a negative parabola?
2. What is the significance of the $x$-intercepts in solving inequalities like $f(x) \geq 0$?
3. How can you verify the solutions to $f(x) = 0$ algebraically if the function is known?
4. What are the different types of solutions for $f(x) \leq 0$?
5. How can you apply this concept to real-world problems involving inequalities?

Tip: When solving inequalities graphically, always look for where the function is on or above (or below) the $x$-axis depending on the inequality sign.