How to determine the distribution function of Y using the density function of X?

[nhrock3]

the first part of the solution finds C
by using the property that the integral on the density function equals 1

the problem starts in the second part of the question
i need to find the distribution function of Y
where Y=||x|-1|
(my problem i written bellow the photo)

for y<0 we have no graph so the probability is 0.
but when its from zero to 1 it touches four points
i don't know how they got the integral from that fact?
the same goes for when y>1
what is the logic?

 

[lippyka]

The logic is that since the density function of X is given, we can use the definition of a distribution function to determine the distribution function of Y. The distribution function of Y is defined as the probability that Y is less than or equal to a given value y. For y<0, the probability that Y is less than or equal to y is 0 since the graph of Y does not intersect the x-axis in this range.For 0<y<1, the probability that Y is less than or equal to y is the area under the graph of the density function of X from -1 to y. This is because the graph of Y intersects the x-axis at x = -1 and x = y in this range, and the area under the graph of the density function of X between these two points gives us the probability that Y is less than or equal to y.For y > 1, the probability that Y is less than or equal to y is the area under the graph of the density function of X from -1 to 1 plus the area under the graph of the density function of X from 1 to y. This is because the graph of Y intersects the x-axis at x = -1 and x = 1 in this range, and the area under the graph of the density function of X between these two points gives us the probability that Y is less than or equal to 1, while the area under the graph of the density function of X from 1 to y gives us the additional probability that Y is less than or equal to y.