@Dulqar , In the solution , Suraj sir has defined a new function 'g' as g(y) = f(y) - f(y+1)
Now In the question , It is given that f(x) is continuous when 0 <= x <= 2 ... and f(0) = f(-2) = -1 and f(1) = 1
Now do the same thing for g(y) = f(y) - f(y+1) by changing variable from x to y and take y from 0 to 1..
Now for g(y)
If , y = 0 , then f(0) = -1 and f(0+1) = f(1) = 1 (given)
so , g(0) = f(0) - f(1) = -1-1 = -2.. so g(0) = -2
If y =1 , then f(1) = 1 and f(1+1)=f(2) = -1 (given)
so , g(1) = f(1) - f(2) = 1 - (-1) = 2..
Now , finally we get g(0) = -2 and g(1) = 2..
Since function is changing its value from -2 (ie. negative ) to +2 (i.e. positive).. It means this function g is increasing and will cut the axis when g(y) = 0..(think same as f(x) when u plot the graph)..
So, g(y) = 0 means f(y) = f(y+1) for the given interval 0 <= y <= 1..
do the same thing for other options.. we will get option A & D