The image presents two functions, $f(x)$ (blue parabola) and $g(x)$ (red line), and the task is to solve two inequalities based on their intersections.

(a) Solving $g(x) \geq f(x)$:

• Identify points of intersection: The functions intersect at $x = 0$ and $x = 4$.
• Behavior between intersections:
  • For $x \leq 0$, the red line $g(x)$ is above the blue curve $f(x)$, satisfying $g(x) \geq f(x)$.
  • For $0 \leq x \leq 4$, $g(x)$ is below $f(x)$, so it does not satisfy the inequality.
  • For $x \geq 4$, $g(x) \geq f(x)$ again holds.

Thus, the solution to $g(x) \geq f(x)$ is $(-\infty, 0] \cup [4, \infty)$.

Would you like a breakdown of part (b) or further clarification?

Here are 5 related questions to expand your understanding:

1. How do you determine where two functions intersect algebraically?
2. What is the difference between solving $f(x) > g(x)$ and $f(x) \geq g(x)$?
3. How does the behavior of a parabola differ from a linear function?
4. How would the inequality change if $f(x)$ and $g(x)$ switched roles?
5. How can the solution to an inequality be verified by substitution?

Tip: Always double-check intersections by plugging the values back into the original functions.