The problem is divided into three exercises. Exercise 1 involves points in an orthonormal coordinate system and asks to find coordinates of vectors, equations of lines, and relationships between points and lines. Exercise 2 concerns a linear function $f$ such that $f(7) = 14$ and asks to find the expression for $f(x)$, find the antecedent of 21 by $f$ and draw the graph of the function $f$. Exercise 3 deals with an affine function $g$ such that $g(0) = 3$ and $g(2) = 7$ and asks to find the expression for $g(x)$, calculate $g(1)$ and $g(3)$, find the antecedent of 21 by $g$ and draw the graph of the function $g$.

1. Problem Description

The problem is divided into three exercises.

Exercise 1 involves points in an orthonormal coordinate system and asks to find coordinates of vectors, equations of lines, and relationships between points and lines.

Exercise 2 concerns a linear function $f$ such that $f(7) = 14$ and asks to find the expression for $f(x)$, find the antecedent of 21 by $f$ and draw the graph of the function $f$.

Exercise 3 deals with an affine function $g$ such that $g(0) = 3$ and $g(2) = 7$ and asks to find the expression for $g(x)$, calculate $g(1)$ and $g(3)$, find the antecedent of 21 by $g$ and draw the graph of the function $g$.

2. Solution Steps

Exercise 1:

1. Plot the points A(1,3), B(2,5), and C(-2,1) on a coordinate plane.

2. Determine the coordinates of vector $AB$:

$AB = (x_B - x_A, y_B - y_A) = (2-1, 5-3) = (1, 2)$

3. Calculate the distance $AB$:

$AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(2-1)^2 + (5-3)^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$

4. Determine the coordinates of the midpoint $I$ of segment $[AB]$:

$I = (\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}) = (\frac{1+2}{2}, \frac{3+5}{2}) = (\frac{3}{2}, \frac{8}{2}) = (\frac{3}{2}, 4)$

5. Show that the reduced equation of line $(AB)$ is $y = 2x + 1$.

The slope of line $(AB)$ is $m = \frac{y_B - y_A}{x_B - x_A} = \frac{5-3}{2-1} = \frac{2}{1} = 2$.

The equation of line $(AB)$ is $y = 2x + b$.

Substituting the coordinates of point $A(1,3)$ into the equation:

$3 = 2(1) + b$

$3 = 2 + b$

$b = 1$

So the equation of line $(AB)$ is $y = 2x + 1$.

6. Does point $C(-2, 1)$ belong to line $(AB)$?

Substituting the coordinates of point $C$ into the equation of line $(AB)$:

$1 = 2(-2) + 1$

$1 = -4 + 1$

$1 = -3$

Since the equation is not satisfied, point $C$ does not belong to line $(AB)$.

7. Determine the reduced equation of line $(D)$ perpendicular to $(AB)$ and passing through point $C$.

The slope of line $(AB)$ is

2. The slope of a line perpendicular to $(AB)$ is $-\frac{1}{2}$.

The equation of line $(D)$ is $y = -\frac{1}{2}x + b$.

Substituting the coordinates of point $C(-2, 1)$ into the equation:

$1 = -\frac{1}{2}(-2) + b$

$1 = 1 + b$

$b = 0$

So the equation of line $(D)$ is $y = -\frac{1}{2}x$.

8. Determine the reduced equation of line $(\Delta)$ parallel to $(AB)$ and passing through point $E(2, 5)$.

The slope of line $(AB)$ is

2. The slope of a line parallel to $(AB)$ is also

2. The equation of line $(\Delta)$ is $y = 2x + b$.

Substituting the coordinates of point $E(2, 5)$ into the equation:

$5 = 2(2) + b$

$5 = 4 + b$

$b = 1$

So the equation of line $(\Delta)$ is $y = 2x + 1$.

Exercise 2:

1. Determine the expression of $f(x)$ given that $f(7) = 14$ and $f$ is a linear function.

Since $f$ is a linear function, $f(x) = ax$.

$f(7) = a(7) = 14$

$a = \frac{14}{7} = 2$

So, $f(x) = 2x$.

2. What is the number $a$ such that $f(a) = 21$?

$f(a) = 2a = 21$

$a = \frac{21}{2} = 10.5$

3. Draw the line $(D)$ representing the graph of the function $f$.

Since $f(x) = 2x$, the line passes through the origin $(0, 0)$ and the point $(1, 2)$.

Exercise 3:

1. Determine the expression of $g(x)$ given that $g(0) = 3$ and $g(2) = 7$ and $g$ is an affine function.

Since $g$ is an affine function, $g(x) = ax + b$.

$g(0) = a(0) + b = 3$, so $b = 3$.

$g(2) = a(2) + 3 = 7$

$2a = 4$

$a = 2$

So, $g(x) = 2x + 3$.

2. Calculate $g(1)$ and $g(3)$:

$g(1) = 2(1) + 3 = 2 + 3 = 5$

$g(3) = 2(3) + 3 = 6 + 3 = 9$

3. What is the number $a$ such that $g(a) = 21$?

$g(a) = 2a + 3 = 21$

$2a = 18$

$a = 9$

4. Draw the line $(\Delta)$ representing the graph of the function $g$.

Since $g(x) = 2x + 3$, the line passes through the points $(0, 3)$ and $(1, 5)$.

3. Final Answer

Exercise 1:

1. Plot the points A(1,3), B(2,5), and C(-2,1).

2. $AB = (1, 2)$

3. $AB = \sqrt{5}$

4. $I = (\frac{3}{2}, 4)$

5. $y = 2x + 1$

6. No, C does not belong to (AB).

7. $y = -\frac{1}{2}x$

8. $y = 2x + 1$

Exercise 2:

1. $f(x) = 2x$

2. $a = 10.5$

3. Draw the line passing through (0,0) and (1,2).

Exercise 3:

1. $g(x) = 2x + 3$

2. $g(1) = 5$ and $g(3) = 9$

3. $a = 9$

4. Draw the line passing through (0,3) and (1,5).