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The problem in this exercise involves two functions \( f \) and \( g \), represented by their respective curves \( C_f \) and \( C_g \) on a coordinate plane. Here’s a breakdown of each part of the exercise:

### 1. Determine \( D_g \) and \( D_f \)
To find the domains \( D_f \) and \( D_g \):
- **\( D_f \)** represents the domain of function \( f \), which includes all \( x \)-values where \( C_f \) is defined.
- **\( D_g \)** represents the domain of function \( g \), which includes all \( x \)-values where \( C_g \) is defined.

Looking at the graph, we observe that:
- The green curve \( C_f \) (representing \( f \)) extends from \( x = 0 \) to the right, indicating \( D_f = [0, +\infty) \).
- The brown curve \( C_g \) (representing \( g \)) spans from \( x = -1.5 \) to \( x = 0 \), indicating \( D_g = [-1.5, 0] \).

### 2. Solve graphically for:
   - **\( f(x) = g(x) \)**: Find the points where \( C_f \) and \( C_g \) intersect.
   - **\( f(x) < g(x) \)**: Determine the intervals where \( C_f \) lies below \( C_g \).
   - **\( f(x) \geq g(x) \)**: Identify the intervals where \( C_f \) lies on or above \( C_g \).
   - **\( g(x) = 0 \)**: Find the points where \( C_g \) intersects the x-axis.
   - **\( g(x) < 0 \)**: Determine where \( C_g \) lies below the x-axis.

#### Observations from the Graph:
1. **\( f(x) = g(x) \)** occurs at two points of intersection:
   - \( x = -1 \)
   - \( x = 0 \)

2. **\( f(x) < g(x) \)** in the interval:
   - \( -1.5 < x < -1 \)

3. **\( f(x) \geq g(x) \)** in the intervals:
   - \( x = -1 \)
   - \( x = 0 \)
   - \( x > 0 \)

4. **\( g(x) = 0 \)** at the point:
   - \( x = 0 \)

5. **\( g(x) < 0 \)** in the interval:
   - \( -1.5 < x < 0 \)

### 3. Determine graphically:
   - **\( f([0, +\infty)) \)**: The range of \( f \) for \( x \in [0, +\infty) \).
   - **\( g([-1.5, 0]) \)**: The range of \( g \) for \( x \in [-1.5, 0] \).

#### Range Analysis:
1. For **\( f([0, +\infty)) \)**, observe the values of \( f \) as \( x \) increases from 0. The range appears to be **\( [0, +\infty) \)**.

2. For **\( g([-1.5, 0]) \)**, observe the values of \( g \) as \( x \) moves from \( -1.5 \) to 0. The range appears to be **\( [-3, 0] \)**.

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Would you like further explanations or have questions about any part? Here are some related questions to deepen understanding:

1. How would the solution change if the domains of \( f \) and \( g \) were different?
2. Why is it important to find the points where \( f(x) = g(x) \)?
3. How can you determine the behavior of \( f(x) \) and \( g(x) \) using derivatives?
4. How would the graph change if \( f(x) \) were shifted upwards or downwards?
5. What happens to the intervals of inequality if \( f \) and \( g \) intersect at more points?

**Tip:** To solve graphical problems, focus on intersection points and relative positions of curves to understand inequalities and ranges effectively.

Soient f et g deux fonctions numérique et (Cf) et (Cg) ses courbes dans un repère orthonormé (O, i, j). 1) Déterminer Dg et Df. 2) Résoudre graphiquement: f(x) = g(x); f(x) < g(x); f(x) ≥ g(x); g(x) = 0; g(x) < 0. 3) Déterminer graphiquement: f([0;+∞[); g([-1.5;0[).

Learn how to determine the domains of functions f and g, analyze their intersection points, and solve inequalities using their graphical representations. This problem covers function domains, graphical inequalities, and range determination in a high school math context.

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