https://file.aiquickdraw.com/mathbot/file/1727588655535679odldk.jpg

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?

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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.

Use the figure to solve the inequalities (a) g(x) ≥ f(x) and (b) f(x) > g(x).

Learn how to solve the inequalities between two functions, f(x) and g(x), by analyzing their intersection points and behavior. Suitable for students in Grades 9-12.

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