Statement (1) is well known as right invariance and statement (2) is known as left invariance..
Now as far as extended transition function is concerned , right invariance holds but not left invariance..
We can think about the same intuitively..
Let we start from start state q0....Now on reading string 'x' and hence applying multiple transition δ*(q0 , x) we reach say state qi...Now given δ*(q0 , x) = δ*(q0 , y)..
So on reading y also we will reach the same state qi....
Hence δ*(q0 , xz) means reading string 'z' after string 'x' will lead to same state as δ*(q0 , yz) as before reading 'z' string in both cases we were in qi state hence the destination state will also be the same.
Hence right invariance holds..
On the other hand ,
for δ*(q0 , zx) and δ*(q0 , zy) , say on reading string 'z' , we reach qi ..After that we read 'x' string for first extended transition function and 'y' string for second one..
Now from qi , on reading 'x' and 'y' we may go to different states..[ As qi is distinct from q0 ]
So even though δ*(q0 , x) = δ*(q0 , y) ,
but δ*(q0 , zx) = δ*(q0 , zy) need not hold...
Hence left invariance does not hold here..
Hence option A) should be correct answer..