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Pachacha [2.7K]
3 years ago
6

The Identity function i behaves just like the number 1. That is (F o i)=f and (i o g)= g

Mathematics
1 answer:
andrezito [222]3 years ago
3 0

I'll denote the identity function by \mathrm{Id}. Then for any functions f with inverse f^{-1},

\begin{cases}f\circ\mathrm{Id}=f\\\mathrm{Id}\circ f=f\\f\circ f^{-1}=\mathrm{Id}\end{cases}

One important fact is that composition is associative, meaning for functions f,g,h, we have

(f\circ g)\circ h=f\circ(g\circ h)

So given

h=f\circ g

we can compose the functions on either side with g^{-1}:

h\circ g^{-1}=(f\circ g)\circ g^{-1}=f\circ(g\circ g^{-1})

then apply the rules listed above:

h\circ g^{-1}=f\circ\mathrm{Id}=f

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