To solve the inequality $f(x) \leq 0$ based on the graph provided, let's follow these steps:

1. Identify the regions where $f(x) \leq 0$:

   • From the graph, $f(x)$ is the y-value of the curve, which is below or equal to the x-axis where $f(x) \leq 0$.
   • Observing the graph, $f(x) \leq 0$ occurs between the points where the graph crosses the x-axis.

2. Determine the x-intercepts:

   • The graph appears to cross the x-axis at approximately $x = -4$ and $x = 2$.

3. Set up the interval where $f(x) \leq 0$:

   • The function $f(x)$ is less than or equal to zero on the interval from $x = -4$ to $x = 2$, inclusive of these endpoints (since $f(x) = 0$ at these points).

4. Express the solution in interval notation:

   • The solution to the inequality $f(x) \leq 0$ is: $[-4, 2]$

Thus, the answer in interval notation is $-4, 2$.

Would you like further details or have any questions?

Related Questions:

1. How do we solve inequalities where $f(x) \geq 0$ instead?
2. What is the process for finding the vertex of a quadratic function from a graph?
3. How can we determine if a function is increasing or decreasing on certain intervals by looking at its graph?
4. What changes in the inequality solution if the function were shifted vertically?
5. How do we solve quadratic inequalities algebraically instead of graphically?

Tip:

Always look for where the graph crosses the x-axis when solving inequalities like $f(x) \leq 0$, as these points often help define the solution intervals.