The given graph represents a quadratic function $y = f(x)$, and the problem asks for the solution set of the inequality $f(x) \leq 0$. Let us analyze the graph and solve:

Steps to Solve:

1. Observe the graph: The quadratic graph opens upwards, with its vertex at $(2, -1)$. It crosses the x-axis at two points: $x = 1$ and $x = 3$.
2. Where is $f(x) \leq 0$?
   • $f(x) \leq 0$ means the function values are either zero or negative. This corresponds to the portion of the graph at or below the x-axis.
   • From the graph, $f(x) \leq 0$ occurs between $x = 1$ and $x = 3$, inclusive (since $f(1) = 0$ and $f(3) = 0$).
3. Solution set: The solution set is the interval $x \in [1, 3]$.

Answer:

The solution set for $f(x) \leq 0$ is not explicitly shown in the given multiple choices. It should ideally be $x \in [1, 3]$. If you’d like, I can verify further.

Would you like more explanation or verification?

Related Questions:

1. How do you find the roots of a quadratic function graphically and algebraically?
2. Why does the interval of $f(x) \leq 0$ only include the x-values between the roots?
3. What is the vertex form of a quadratic function, and how is it used?
4. Can we write the solution set using inequalities rather than intervals?
5. How does the direction of the parabola (upwards or downwards) affect solutions to inequalities?

Tip:

Always verify the direction of the parabola (up or down) when solving quadratic inequalities. This determines where $f(x)$ is positive or negative.