Let $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ be two invertible functions. Then prove that $gof$ is also invertible with $(gof)^{-1}=f^{-1}og^{-1}$.

CBSE Class 12 · 2005 · Relations And Functions XIILong Answer

Let

f : X \to Y

and

g : Y \to Z

be two invertible functions. Then prove that

g o f

is also invertible with

(g o f)^{- 1} = f^{- 1} o g^{- 1}

Answer: $g o f$ is invertible and $(g o f)^{- 1} = f^{- 1} o g^{- 1}$ .

Since

f

and

g

are invertible, they are bijections; the composition of bijections is a bijection, so

g o f : X \to Z

is invertible.
It remains to identify its inverse. Let

h = f^{- 1} o g^{- 1} : Z \to X

and check both compositions give identities.
Compose

g o f

with

h

on one side:

(g o f) o (f^{- 1} o g^{- 1}) = g o (f o f^{- 1}) o g^{- 1} = g o I_{Y} o g^{- 1} = g o g^{- 1} = I_{Z}

Now the other side:

(f^{- 1} o g^{- 1}) o (g o f) = f^{- 1} o (g^{- 1} o g) o f = f^{- 1} o I_{Y} o f = f^{- 1} o f = I_{X}

Both compositions are identities, so

h

is the two-sided inverse of

g o f

(g o f)^{- 1} = f^{- 1} o g^{- 1}

CBSE Class 12 Board · 2005 · Subjective