https://file.aiquickdraw.com/mathbot/file/17267738734414k2r3pyv.jpg

Let's work through the given questions based on the graph of the functions \( f(x) \) (red curve) and \( g(x) \) (blue curve).

### (a) Which is larger, \( f(0) \) or \( g(0) \)?
- **From the graph**, at \( x = 0 \), the red curve \( f(x) \) is at \( y = 2 \), and the blue curve \( g(x) \) is at \( y = 0 \).
- **Answer**: \( f(0) \) is larger because \( f(0) = 2 \) and \( g(0) = 0 \).

### (b) Which is larger, \( f(-3) \) or \( g(-3) \)?
- **From the graph**, at \( x = -3 \), \( f(x) \) appears to be at around \( y = 1 \), while \( g(x) \) is at a negative value, around \( y = -2 \).
- **Answer**: \( f(-3) \) is larger because \( f(-3) \approx 1 \) and \( g(-3) \approx -2 \).

### (c) For which values of \( x \) is \( f(x) = g(x) \)?
- **From the graph**, the two curves intersect at two points: around \( x = -2 \) and \( x = 2 \).
- **Answer**: \( f(x) = g(x) \) at \( x = -2 \) and \( x = 2 \).

### (d) Find the values of \( x \) for which \( f(x) \leq g(x) \).
- \( f(x) \) is less than or equal to \( g(x) \) where the red curve is either below or touching the blue curve.
- **From the graph**, this happens in the intervals \( (-\infty, -2] \cup [2, \infty) \).
- **Answer**: \( f(x) \leq g(x) \) for \( x \in (-\infty, -2] \cup [2, \infty) \).

### (e) Find the values of \( x \) for which \( f(x) > g(x) \).
- \( f(x) \) is greater than \( g(x) \) where the red curve is above the blue curve.
- **From the graph**, this happens between \( x = -2 \) and \( x = 2 \), not including those points.
- **Answer**: \( f(x) > g(x) \) for \( x \in (-2, 2) \).

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Would you like more details on any of these answers? Here are some related questions for further practice:

1. What is the behavior of \( f(x) \) and \( g(x) \) as \( x \to \infty \)?
2. At what points does the slope of \( f(x) \) equal zero?
3. What is the maximum value of \( g(x) \) and at what \( x \)-value does it occur?
4. Where does the function \( g(x) \) have a minimum value?
5. Can you determine whether the functions are increasing or decreasing at \( x = 1 \)?

**Tip**: When solving graphically, always focus on the relative positions of the curves to compare their values at specific points.

Graphs of the functions f and g are given. (a) Which is larger, f(0) or g(0)? (b) Which is larger, f(-3) or g(-3)? (c) For which values of x is f(x) = g(x)? (d) Find the values of x for which f(x) ≤ g(x). (e) Find the values of x for which f(x) > g(x).

Explore how to compare functions f and g graphically, solving inequalities, and finding the points where f(x) equals g(x). This problem covers essential graph interpretation skills and function comparison for high school students.

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