# If g o f = idA and f o g = idB, then g = f^-1 proof?

• Norm850
In summary, the conversation discusses the proof that if two sets A and B have functions f and g with f: A -> B and g: B -> A, and g o f = idA and f o g = idB, then f is invertible and g = f^-1. The conversation also includes a hint for proving this, and a question about the necessity for f to be 1-1 and onto. It is determined that the proof is sufficient for g = f^-1, but it must first be shown that f^-1 exists.
Norm850
Hey guys,

I need to prove: Suppose A and B are sets, and f and g are functions with f: A -> B and g: B -> A. If g o f = idA and f o g = idB, then f is invertible and g = f^-1.

So far I have understood why g must be the inverse of f, but I do not know how to prove it.

Thanks!

Take an arbitrary element a in A and b in B and show that they relate via composition of both functions.

Hint: f(a) = b, g(b) = a

tazzzdo said:
Take an arbitrary element a in A and b in B and show that they relate via composition of both functions.

Okay, not really sure how to do that but here's my attempt:

Suppose g o f = idA,
then dom(g o f) = dom(idA) = A
and (g o f)(a) = g(f(a)) = idA(a) = a.

Suppose f o g = idB,
then dom(f o g) = dom(idB) = B
and (f o g)(b) = f(g(b)) = idB(b) = b.

A function is only invertible if it's 1 - 1 and onto. Is this the case? You didn't specify.

tazzzdo said:
A function is only invertible if it's 1 - 1 and onto. Is this the case? You didn't specify.

So what I have is sufficient for g=f^-1? But first I need to show that f^-1 exists?

## 1. What does "g o f = idA" mean?

The notation "g o f" represents the composition of two functions, g and f, where the output of f becomes the input for g. The "idA" signifies the identity function, which returns the same value as its input. Therefore, "g o f = idA" means that the composition of g and f results in the identity function for the domain A.

## 2. How is this related to the inverse of a function?

The inverse of a function undoes the original function's operation. In this context, "f o g = idB" implies that the composition of f and g results in the identity function for the domain B. Since the composite functions result in identity functions for different domains, it can be concluded that g is the inverse of f and vice versa.

## 3. What does the notation "g = f^-1" mean?

The notation "g = f^-1" represents the inverse function of f, which undoes f's operation. This is equivalent to swapping the input and output of the original function f.

## 4. How can we prove that g is the inverse of f?

To prove that g is the inverse of f, we can use the fact that "g o f = idA" and "f o g = idB". These conditions satisfy the definition of inverse functions, where the composition of a function with its inverse results in the identity function.

## 5. Can this proof be applied to all functions?

Yes, this proof can be applied to all functions as long as the composition of g and f results in the identity function for the respective domains. However, it is important to note that not all functions have inverses, as some may not have a one-to-one correspondence between their input and output values.

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