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

In summary, the conversation discusses whether or not f o g is a function and if (f o g)^-1 = gof, with the conclusion being that the first part is true but the second part is false. It is also mentioned that (gof)^-1 actually equals f^-1og^-1, and that it is a function from C to A.
  • #1
Monsu
38
1
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
 
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  • #2
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.
 
  • #3
fog is not a function, composition reads from right to left. gof is a function.
 
  • #4
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.
 

1. What is a composite function?

A composite function is a combination of two or more functions where the output of one function is used as the input for another function. This is denoted as f o g, with f and g being the individual functions.

2. How do you find the inverse of a composite function?

To find the inverse of a composite function (f o g)^-1, you need to first find the inverse of each individual function. Then, you can switch the order of the functions and combine them to get gof.

3. Is (f o g)^-1 always equal to gof?

No, (f o g)^-1 is only equal to gof if both f and g are invertible functions. If one or both of the functions are not invertible, then (f o g)^-1 and gof will not be equal.

4. How can I prove that (f o g)^-1 = gof?

To prove that (f o g)^-1 = gof, you can use the definition of inverse functions. This involves showing that (f o g)(gof(x)) = x and (g o f)(fog(x)) = x for all values of x in the domains of f and g.

5. Can the order of the functions in a composite function be changed?

Yes, the order of functions in a composite function can be changed. However, this will only result in the same output if both functions are commutative, meaning that the order of the inputs and outputs does not affect the result. Otherwise, changing the order of the functions will result in a different composite function.

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