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The image shows a mathematical activity involving a function \( f \) defined on \( \mathbb{R} \) by the expression:

\[
f(x) = 2 |x - 2| + |x + 3| - x^2
\]

The activity has two tasks:

1. **Donner l'expression de la fonction \( g \)**: This asks for the expression of the function \( g \), which is likely a restriction of \( f \) over a specific interval, as mentioned next.
   
2. **Représenter graphiquement \( g \)**: This asks for a graphical representation of the function \( g \).

Additionally, \( g \) is defined as the restriction of \( f \) over the interval \([2, +\infty[\).

Let’s break this down:

### 1. **Expression de la fonction \( g \)**:
- Since \( g \) is the restriction of \( f \) to the interval \([2, +\infty[\), we need to analyze the behavior of the absolute value terms \( |x - 2| \) and \( |x + 3| \) on this interval.

- For \( x \geq 2 \):
   - \( |x - 2| = x - 2 \) (since \( x - 2 \geq 0 \)),
   - \( |x + 3| = x + 3 \) (since \( x + 3 \geq 0 \) for all \( x \geq 2 \)).
   
Thus, the function \( g(x) \) on the interval \( [2, +\infty[ \) simplifies to:

\[
g(x) = 2(x - 2) + (x + 3) - x^2
\]
Simplifying further:
\[
g(x) = 2x - 4 + x + 3 - x^2 = 3x - 1 - x^2
\]
So, the expression of \( g(x) \) is:

\[
g(x) = -x^2 + 3x - 1
\]

### 2. **Représentation graphique**:
To graph \( g(x) = -x^2 + 3x - 1 \), you would plot a parabola that opens downward (because of the \( -x^2 \) term), with its vertex at \( x = \frac{-b}{2a} = \frac{-3}{2(-1)} = 1.5 \), and the parabola is restricted to \( x \geq 2 \).

Would you like to dive into more details or see the graph plotted?

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Here are 5 related questions to expand on this topic:

1. How does the vertex form of a quadratic equation help in graphing the function \( g(x) \)?
2. What are the roots of the quadratic function \( g(x) = -x^2 + 3x - 1 \)?
3. How does restricting the domain affect the behavior of the function \( f(x) \)?
4. Can the function \( g(x) \) have any critical points in its domain \( [2, +\infty[ \)?
5. What is the significance of the absolute value terms in the original function \( f(x) \)?

**Tip**: When graphing quadratic functions, always find the vertex and direction of the parabola (upward or downward) for better accuracy!

On considère la fonction définie par f(x) = 2|x - 2| + |x + 3| - x^2. Donner l'expression de la fonction g, restriction de f à l'intervalle [2, +∞[, et représenter graphiquement g.

Découvrez comment obtenir l'expression de la fonction g, une restriction de f(x) = 2|x - 2| + |x + 3| - x^2 sur l'intervalle [2, +∞[, et apprenez à la représenter graphiquement. Ce problème est adapté aux élèves de la 2nde à la terminale.

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