Answer

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Hint: At first we have to check if the functions $f,g$ are one to one and onto or not. A function is invertible only if the function is both one to one and onto.

Complete step-by-step answer:

The function f is defined as:

$f:\left\{ 1,2,3 \right\}\to \left\{ a,b,c \right\}$.

$f(1)=a,f(2)=b,f(3)=c$ .

Here $\left\{ 1,2,3 \right\}$ is the domain of the function f. $\left\{ a,b,c \right\}$ is the codomain of the function f.

We know that a function is said to be one to one if every different element of the domain has different images.

Here image of 1 is a. Image of 2 is b. Image of 3 is c. Therefore, every different element of the domain has a different image. Hence, f is one to one.

We know that a function is said to be onto if for every element of the codomain, we can find out at least one preimage from the domain.

The preimage of a is 1. Preimage of b is 2. Preimage of c is 3.

Therefore, every element of the codomain has a preimage. Hence, f is a onto function.

Therefore, f is both one to one and onto. So, f is invertible.

Similarly, the function g is defined as:

$g:\left\{ a,b,c \right\}\to \left\{ apple,ball,cat \right\}$

$g(a)=apple, g(b)=ball, g(c)=cat$

Here $\left\{ a,b,c \right\}$ is the domain of the function. $\left\{ apple,ball,cat \right\}$ is the codomain of the function.

The function g is one to one as image of a is apple, image of b is ball, image of c is cat. Therefore every element of the domain has a different image.

The function g is onto as preimage of apple is a, preimage of ball is b, preimage of cat is c. Therefore every element of the codomain has one preimage in the domain.

Hence, g is both one to one and onto. So, g is invertible.

Now, $(g\circ f):\left\{ 1,2,3 \right\}\to \left\{ apple,ball,cat \right\}$ is defined as:

As $\left( g\circ f \right)\left( 1 \right)=g\left( f\left( 1 \right) \right)=g\left( a \right)=apple$

$\begin{align}

& (g\circ f)(2)=g\left( f\left( 2 \right) \right)=g(b)=ball \\

& (g\circ f)(3)=g\left( f\left( 3 \right) \right)=g(c)=cat \\

\end{align}$

$g\circ f$ is one to one as every element of the domain $\left\{ 1,2,3 \right\}$ has different image.

$g\circ f$ is onto as every element of the codomain $\left\{ apple,ball,cat \right\}$ has a preimage in the domain.

Therefore $g\circ f$ is invertible.

We know that if a function $f$ maps one element $x$ to $y$, then the inverse function maps the image $y$ to $x$. That is:

$f\left( x \right)=y\Rightarrow {{f}^{-1}}\left( y \right)=x$

Therefore,

$\begin{align}

& f(1)=a\Rightarrow {{f}^{-1}}\left( a \right)=1 \\

& f(2)=b\Rightarrow {{f}^{-1}}\left( b \right)=2 \\

& f(3)=c\Rightarrow {{f}^{-1}}\left( c \right)=3 \\

\end{align}$

Hence,

${{f}^{-1}}:\left\{ a,b,c \right\}\to \left\{ 1,2,3 \right\}$ , such that:

${{f}^{-1}}\left( a \right)=1,{{f}^{-1}}\left( b \right)=2,{{f}^{-1}}\left( c \right)=3$

Similarly,

$\begin{align}

&g\left( a \right)=apple\Rightarrow {{g}^{-1}}\left( apple \right)=a \\

&g\left( b \right)=ball\Rightarrow {{g}^{-1}}\left( ball \right)=b \\

&g\left( c \right)=cat\Rightarrow {{g}^{-1}}\left( cat \right)=c \\

\end{align}$

Therefore,

${{g}^{-1}}:\left\{ apple,ball,cat \right\}\to \left\{ a,b,c \right\}$ , such that:

$\begin{align}

& {{g}^{-1}}(apple)=a \\

& {{g}^{-1}}\left( ball \right)=b \\

& {{g}^{-1}}\left( cat \right)=c \\

\end{align}$

Similarly,

$\begin{align}

& \left( g\circ f \right)\left( 1 \right)=apple\Rightarrow {{\left( g\circ f \right)}^{-1}}\left( apple \right)=1 \\

& \left( g\circ f \right)\left( 2 \right)=ball\Rightarrow {{\left( g\circ f \right)}^{-1}}\left( ball \right)=2 \\

& \left( g\circ f \right)\left( 3 \right)=cat\Rightarrow {{\left( g\circ f \right)}^{-1}}\left( cat \right)=3 \\

\end{align}$

Therefore,

${{\left( g\circ f \right)}^{-1}}:\left\{ apple,ball,cat \right\}\to \left\{ 1,2,3 \right\}$ , such that:

$\begin{align}

& {{\left( g\circ f \right)}^{-1}}\left( apple \right)=1 \\

& {{\left( g\circ f \right)}^{-1}}\left( ball \right)=b \\

& {{\left( g\circ f \right)}^{-1}}\left( cat \right)=c \\

\end{align}$

Now,

$\begin{align}

& \left( {{f}^{-1}}\circ {{g}^{-1}} \right)\left( apple \right)={{f}^{-1}}\left( {{g}^{-1}}\left( apple \right) \right)={{f}^{-1}}\left( a \right)=1 \\

& \left( {{f}^{-1}}\circ {{g}^{-1}} \right)\left( ball \right)={{f}^{-1}}\left( {{g}^{-1}}\left( ball \right) \right)={{f}^{-1}}\left( b \right)=2 \\

& \left( {{f}^{-1}}\circ {{g}^{-1}} \right)\left( cat \right)={{f}^{-1}}\left( {{g}^{-1}}\left( cat \right) \right)={{f}^{-1}}\left( c \right)=3 \\

& \\

\end{align}$

Therefore,

$\begin{align}

& {{\left( g\circ f \right)}^{-1}}\left( apple \right)=\left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( apple \right) \\

& {{\left( g\circ f \right)}^{-1}}\left( ball \right)=\left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( ball \right) \\

& {{\left( g\circ f \right)}^{-1}}\left( cat \right)=\left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( cat \right) \\

\end{align}$

Hence, ${{\left( g\circ f \right)}^{-1}}=\left( {{f}^{-1}}\circ {{g}^{-1}} \right)$

Note: We generally make mistakes to find out the inverse function. Always remember:

$f\left( x \right)=y\Rightarrow {{f}^{-1}}\left( y \right)=x$

Complete step-by-step answer:

The function f is defined as:

$f:\left\{ 1,2,3 \right\}\to \left\{ a,b,c \right\}$.

$f(1)=a,f(2)=b,f(3)=c$ .

Here $\left\{ 1,2,3 \right\}$ is the domain of the function f. $\left\{ a,b,c \right\}$ is the codomain of the function f.

We know that a function is said to be one to one if every different element of the domain has different images.

Here image of 1 is a. Image of 2 is b. Image of 3 is c. Therefore, every different element of the domain has a different image. Hence, f is one to one.

We know that a function is said to be onto if for every element of the codomain, we can find out at least one preimage from the domain.

The preimage of a is 1. Preimage of b is 2. Preimage of c is 3.

Therefore, every element of the codomain has a preimage. Hence, f is a onto function.

Therefore, f is both one to one and onto. So, f is invertible.

Similarly, the function g is defined as:

$g:\left\{ a,b,c \right\}\to \left\{ apple,ball,cat \right\}$

$g(a)=apple, g(b)=ball, g(c)=cat$

Here $\left\{ a,b,c \right\}$ is the domain of the function. $\left\{ apple,ball,cat \right\}$ is the codomain of the function.

The function g is one to one as image of a is apple, image of b is ball, image of c is cat. Therefore every element of the domain has a different image.

The function g is onto as preimage of apple is a, preimage of ball is b, preimage of cat is c. Therefore every element of the codomain has one preimage in the domain.

Hence, g is both one to one and onto. So, g is invertible.

Now, $(g\circ f):\left\{ 1,2,3 \right\}\to \left\{ apple,ball,cat \right\}$ is defined as:

As $\left( g\circ f \right)\left( 1 \right)=g\left( f\left( 1 \right) \right)=g\left( a \right)=apple$

$\begin{align}

& (g\circ f)(2)=g\left( f\left( 2 \right) \right)=g(b)=ball \\

& (g\circ f)(3)=g\left( f\left( 3 \right) \right)=g(c)=cat \\

\end{align}$

$g\circ f$ is one to one as every element of the domain $\left\{ 1,2,3 \right\}$ has different image.

$g\circ f$ is onto as every element of the codomain $\left\{ apple,ball,cat \right\}$ has a preimage in the domain.

Therefore $g\circ f$ is invertible.

We know that if a function $f$ maps one element $x$ to $y$, then the inverse function maps the image $y$ to $x$. That is:

$f\left( x \right)=y\Rightarrow {{f}^{-1}}\left( y \right)=x$

Therefore,

$\begin{align}

& f(1)=a\Rightarrow {{f}^{-1}}\left( a \right)=1 \\

& f(2)=b\Rightarrow {{f}^{-1}}\left( b \right)=2 \\

& f(3)=c\Rightarrow {{f}^{-1}}\left( c \right)=3 \\

\end{align}$

Hence,

${{f}^{-1}}:\left\{ a,b,c \right\}\to \left\{ 1,2,3 \right\}$ , such that:

${{f}^{-1}}\left( a \right)=1,{{f}^{-1}}\left( b \right)=2,{{f}^{-1}}\left( c \right)=3$

Similarly,

$\begin{align}

&g\left( a \right)=apple\Rightarrow {{g}^{-1}}\left( apple \right)=a \\

&g\left( b \right)=ball\Rightarrow {{g}^{-1}}\left( ball \right)=b \\

&g\left( c \right)=cat\Rightarrow {{g}^{-1}}\left( cat \right)=c \\

\end{align}$

Therefore,

${{g}^{-1}}:\left\{ apple,ball,cat \right\}\to \left\{ a,b,c \right\}$ , such that:

$\begin{align}

& {{g}^{-1}}(apple)=a \\

& {{g}^{-1}}\left( ball \right)=b \\

& {{g}^{-1}}\left( cat \right)=c \\

\end{align}$

Similarly,

$\begin{align}

& \left( g\circ f \right)\left( 1 \right)=apple\Rightarrow {{\left( g\circ f \right)}^{-1}}\left( apple \right)=1 \\

& \left( g\circ f \right)\left( 2 \right)=ball\Rightarrow {{\left( g\circ f \right)}^{-1}}\left( ball \right)=2 \\

& \left( g\circ f \right)\left( 3 \right)=cat\Rightarrow {{\left( g\circ f \right)}^{-1}}\left( cat \right)=3 \\

\end{align}$

Therefore,

${{\left( g\circ f \right)}^{-1}}:\left\{ apple,ball,cat \right\}\to \left\{ 1,2,3 \right\}$ , such that:

$\begin{align}

& {{\left( g\circ f \right)}^{-1}}\left( apple \right)=1 \\

& {{\left( g\circ f \right)}^{-1}}\left( ball \right)=b \\

& {{\left( g\circ f \right)}^{-1}}\left( cat \right)=c \\

\end{align}$

Now,

$\begin{align}

& \left( {{f}^{-1}}\circ {{g}^{-1}} \right)\left( apple \right)={{f}^{-1}}\left( {{g}^{-1}}\left( apple \right) \right)={{f}^{-1}}\left( a \right)=1 \\

& \left( {{f}^{-1}}\circ {{g}^{-1}} \right)\left( ball \right)={{f}^{-1}}\left( {{g}^{-1}}\left( ball \right) \right)={{f}^{-1}}\left( b \right)=2 \\

& \left( {{f}^{-1}}\circ {{g}^{-1}} \right)\left( cat \right)={{f}^{-1}}\left( {{g}^{-1}}\left( cat \right) \right)={{f}^{-1}}\left( c \right)=3 \\

& \\

\end{align}$

Therefore,

$\begin{align}

& {{\left( g\circ f \right)}^{-1}}\left( apple \right)=\left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( apple \right) \\

& {{\left( g\circ f \right)}^{-1}}\left( ball \right)=\left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( ball \right) \\

& {{\left( g\circ f \right)}^{-1}}\left( cat \right)=\left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( cat \right) \\

\end{align}$

Hence, ${{\left( g\circ f \right)}^{-1}}=\left( {{f}^{-1}}\circ {{g}^{-1}} \right)$

Note: We generally make mistakes to find out the inverse function. Always remember:

$f\left( x \right)=y\Rightarrow {{f}^{-1}}\left( y \right)=x$

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