# What is the function f(x) which has the graph of the segment AB A(-1,-3), B(3,7) ?

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The segment AB is a line joining the points A( -1 , -3) and B(3, 7).

The equation of the line joining them can be taken to be the function f(x).

The equation of the line is given by:

y + 3 = [( -3 - 7)/(-1 - 3)](x + 1)

=> y + 3 = (-10/-4)(x + 1)

=> y +3 = (5/2)(x + 1)

=> y = (5/2)(x + 1) - 3

=> y = (5x + 5 - 6)/2

=> y = (5x - 1)/2

Therefore **f(x) = (5x - 1)/2**

We'll write the form of the linear function:

f(x) = ax + b

or

y = mx + n, where m represents the slope of the line and n represents the y intercept.

In this case, the function f(x) has as graph the line AB.

According to the rule, a point belongs to a line if the coordinates of the point verify the equation of the line.

A is on the line y = ax+b if and only if yA = a*xA + b

We'll substitute the coordinates xA and yA and we'll get:

-3 = a*(-1) + b

-3 = -a + b (1)

B belongs to the line y = ax+b if and only if yB = a*xB + b.

We'll substitute the coordinates xB and yB and we'll get:

7 = a*3 + b (2)

We'll subtract (2) from (1):

3a + b + a - b = 7 + 3

We'll combine and eliminate like terms:

4a = 10

a = 5/2

b - a = -3

b = a - 3

b = 5/2 - 3

b = -1/2

The function f(x) whose graph is represented by the line AB:

**f(x) = (5/2)*x - 1/2**

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