Composite Functions: Is (f o g)^-1 = gof?

[Monsu]

If f:A -> B and g:B -> C are functions, is this true: f o g is also a function and (f o g) ^-1 = gof

I think this isn't true, but if this isn't the case, could someone please tell me a counter example?? Thanks

 

[modmans2ndcoming]

the first part is true, but the send part is false. since it is an and statement the entire statement is false. if it were an or statement, it would be true.

I think I am right but I can't remember right now if the mapping from A to B gets changed when you take the inverse or what. either way, it should not be a function any more.

 

[matt grime]

fog is not a function, composition reads from right to left. gof is a function.

 

[HallsofIvy]

Assuming f:A->B and g:B->C then gof is a function from A to C.

No, it is NOT true that (gof)^-1= fog. As matt grime pointed out, that's not even a function: g goes from B to C and C is not the domain of f.

What IS true, in this case, is that (gof)^-1= f^-1og^-1 which is probably what you meant.

Notice that, since f:A->B, f^-1:B->A and, since g:B->C, g^-1:C->B. That means that both (gof)^-1 and f^-1og^-1 are from C to A.