I found the following from a book with no proof:

Let \(\displaystyle f:A \rightarrow A\) be defined by the formula \(\displaystyle f(x)=x\), then \(\displaystyle f\) is called the identity function, denoted by \(\displaystyle 1\) or by \(\displaystyle 1_A\).

Let \(\displaystyle f:A \rightarrow B\) and it has the inverse function \(\displaystyle f^{-1}:B\rightarrow A\), then \(\displaystyle f ^{-1}\circ f=1\)

Question: Is it true that \(\displaystyle f ^{-1}\circ f=1\)?

Isn't \(\displaystyle f(x)^{-1} \circ f(x)=x\)?

Let \(\displaystyle f:A \rightarrow A\) be defined by the formula \(\displaystyle f(x)=x\), then \(\displaystyle f\) is called the identity function, denoted by \(\displaystyle 1\) or by \(\displaystyle 1_A\).

Let \(\displaystyle f:A \rightarrow B\) and it has the inverse function \(\displaystyle f^{-1}:B\rightarrow A\), then \(\displaystyle f ^{-1}\circ f=1\)

Question: Is it true that \(\displaystyle f ^{-1}\circ f=1\)?

Isn't \(\displaystyle f(x)^{-1} \circ f(x)=x\)?

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