stuck on this one
a) (fog)(2)=f(g(2))=f(2*2)=f(4)=4+1=5
b) (gof)(2)=g(f(2)) = g(2+1) = g(3) = 2*3=6
Hence $\displaystyle (fog)(2)\neq (gof)(2)$
The function: f={(1,a),(2,a),(4,c),(3,d) from A={1,2,3,4} TO B= {a,b,c,d}
IS not one to one and onto,since
there exist x=1 ,y=2, such that f(x) =f(y)=a, and $\displaystyle x\neq y$
Note for a function to be one to one we must have:
for all x, yεA if f(x)=f(y) ,then x=y
Also for a function to be onto we must have:
for all y belonging to B ,yεB, there exists xεA SUCH that f(x)=y.
However in our case we note that:
there exists y=bεB SUCH that FOR no element of xεA WE have f(x)=y=b
Hence there is no inverse function from B TO A.
This can be seen in another way:
Let the inverse of f be defined as :f^(-1)={(y,x): (x,y)εf},and according to that definition :
f^(-1) = {(a,1),(a,2),(c,4),(d,3)}, and using the definition of a function we observe that f^(-1) ( the inverse of f) cannot be a function from B TO A