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The exercise provides two functions \( f(x) = 2x - 3 \) and \( g(x) = \frac{2 - 3x}{4} \) and requires the following tasks:

### 1. **Graph the curves of \( f \) and \( g \) on an orthonormal coordinate system with a scale of 2 cm per unit.**

Before plotting, let’s analyze each function.

- For \( f(x) = 2x - 3 \):
  - The slope is 2, meaning the line is increasing.
  - The y-intercept is -3, so the line passes through \( (0, -3) \).

- For \( g(x) = \frac{2 - 3x}{4} \):
  - Simplifying this expression: \( g(x) = \frac{2}{4} - \frac{3x}{4} = \frac{1}{2} - \frac{3x}{4} \).
  - This is a linear function with slope \( -\frac{3}{4} \) and y-intercept \( \frac{1}{2} \), meaning the line is decreasing and passes through \( (0, 0.5) \).

### 2. **Determine by calculation the coordinates of the intersection points of the two curves.**

To find the points of intersection, we need to solve the equation where \( f(x) = g(x) \).

So, set:
\[
2x - 3 = \frac{1}{2} - \frac{3x}{4}
\]

Let's solve this equation step by step:

\[
2x - 3 = \frac{1}{2} - \frac{3x}{4}
\]
Multiply the whole equation by 4 to eliminate the fractions:

\[
4(2x - 3) = 4\left(\frac{1}{2} - \frac{3x}{4}\right)
\]

This simplifies to:

\[
8x - 12 = 2 - 3x
\]

Now, add \( 3x \) to both sides to combine the terms with \( x \):

\[
8x + 3x - 12 = 2
\]

Simplify:

\[
11x - 12 = 2
\]

Now, add 12 to both sides:

\[
11x = 14
\]

Finally, divide by 11:

\[
x = \frac{14}{11}
\]

Substitute this value of \( x \) into either \( f(x) \) or \( g(x) \) to find \( y \).

Substitute into \( f(x) = 2x - 3 \):

\[
f\left(\frac{14}{11}\right) = 2 \times \frac{14}{11} - 3 = \frac{28}{11} - 3 = \frac{28}{11} - \frac{33}{11} = \frac{-5}{11}
\]

So, the point of intersection is \( \left( \frac{14}{11}, \frac{-5}{11} \right) \).

---

Would you like further clarification on the graphing part or any other aspect of this problem?

### Relative Questions:
1. How do you find the slope of a line from its equation?
2. How do the slopes of two lines affect their points of intersection?
3. What does the y-intercept of a linear function represent?
4. What is the graphical meaning of solving \( f(x) = g(x) \)?
5. How can you interpret the result of \( x = \frac{14}{11} \) geometrically?

### Tip:
When solving equations involving fractions, multiplying through by the least common denominator helps eliminate the fractions quickly.

Soient f et g les deux fonctions définies sur ℝ par f(x) = 2x - 3 et g(x) = (2 - 3x)/4. Tracer les courbes représentatives de f et g dans un repère orthonormé d’échelle 2cm pour une unité. Déterminer par le calcul les coordonnées des éventuels points d’intersection des deux courbes.

Learn how to graph the linear functions f(x) = 2x - 3 and g(x) = (2 - 3x)/4 and calculate the points of intersection. This problem covers key concepts of graphing and solving linear equations, suitable for students in Grades 9-11.

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