Notice that the empty circle means that the point circled doesn't belong to the range of the function. Otherwise, the fulfilled circle means the point belongs to that range.
The first horizontal line represents the values of the function for x in the interval 0 < x ≤ 15 (notice the empty circle in point (0, 35) ).
Since this line is horizontal, it means that the function has the same value for all those x's:
f(x) = 35, for 0 < x ≤ 15
Now, let's look at the other horizontal line. It represents the values of the function for all x's in the interval 15 < x ≤ 40 (again, notice the empty circle in point (15, 40) ).
Again, since it's a horizontal line, the function has the same value for all those x's:
f(x) = 40, for 15 < x ≤ 40
Finally, the third line has a slope that we have to consider in order to find the form of the function for the last values of x.
We can find the slope in the following way:
1. first, we choose two points in this line whose coordinates are easy for us to identify. Let's take the points (40, 40) and (45, 50).
2. then, we calculate the difference between the y values of these two points:
y2 - y1 = 50 - 40 = 10
3. then, we calculate the difference between the x values of these two points (remember to take the second one minus the first one):
x2 - x1 = 45 - 40 = 5
4. now, we divide the y's differences by the x's differences (this will be the slope m):
m = (y2 - y1)/(x2 - x1) = 10/5 = 2
Now we know the slope m, we write the function of this line in the form:
f(x) - f(x0) = m * (x - x0)
f(x) - f(x0) = 2 * (x - x0)
Finally, to find the final form of this function, we choose any point in the line and use its x value replacing x0, and its y value replacing f(x0). Let's use the point (40, 40):
f(x) - f(x0) = 2 * (x - x0)
f(x) - 40 = 2 * (x - 40)
f(x) - 40 = 2x - 80
f(x) = 2x - 80 + 40
f(x) = 2x -40
Notice that this function for x = 40 has to assume the same value as the one we find for x = 40 using the function for the second horizontal line (f(x) = 40 ), because they both are defined for the same point:
f(x) = 2x -40
f(40) = 2*40 - 40 = 80 - 40 = 40
For this reason, as the second line already defines the function for x = 40, we can write the third part of the function only for values above x > 40.
Therefore, the function that represents the whole graph is:
f(x) = 35, for 0 < x ≤ 15
f(x) = 40, for 15 < x ≤ 40
f(x) = 2x -40, for x > 40