Search any question & find its solution
Question:
Answered & Verified by Expert
Consider $f:\{1,2,3\} \rightarrow\{a, b, c\}$ given by $f(1)=a, f(2)=b$ and $f(3)=c$. Find $f^{-1}$ and show that $\left(f^{-1}\right)^{-1}=f$.
Solution:
1138 Upvotes
Verified Answer
$f:\{1,2,3,\} \rightarrow\{a, b, c\}$ so that $f(1)=a, f(2)=b, f(3)=c$
Now let $\mathrm{X}=\{1,2,3\}, \mathrm{Y}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$
$\therefore \mathrm{f}: \mathrm{X} \rightarrow \mathrm{Y} \quad \therefore \mathrm{f}^{-1}: \mathrm{Y} \rightarrow \mathrm{X}$ such that $\mathrm{f}^{-1}(\mathrm{a})=1$, $\mathrm{f}^{-1}(\mathrm{~b})=2 ; \mathrm{f}^{-1}(\mathrm{c})=3$
Inverse of this function may be written as $\left(\mathrm{f}^{-1}\right)^{-1}: \mathrm{X} \rightarrow \mathrm{Y}$ such that
$$
\left(f^{-1}\right)^{-1}(1)=a,\left(f^{-1}\right)^{-1}(2)=b,\left(f^{-1}\right)^{-1}(3)=c
$$
We also have $f: X \rightarrow Y$ such that
$$
f(1)=a, f(2)=b, f(3)=c \Rightarrow\left(f^{-1}\right)^{-1}=f \text {. }
$$
Now let $\mathrm{X}=\{1,2,3\}, \mathrm{Y}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$
$\therefore \mathrm{f}: \mathrm{X} \rightarrow \mathrm{Y} \quad \therefore \mathrm{f}^{-1}: \mathrm{Y} \rightarrow \mathrm{X}$ such that $\mathrm{f}^{-1}(\mathrm{a})=1$, $\mathrm{f}^{-1}(\mathrm{~b})=2 ; \mathrm{f}^{-1}(\mathrm{c})=3$
Inverse of this function may be written as $\left(\mathrm{f}^{-1}\right)^{-1}: \mathrm{X} \rightarrow \mathrm{Y}$ such that
$$
\left(f^{-1}\right)^{-1}(1)=a,\left(f^{-1}\right)^{-1}(2)=b,\left(f^{-1}\right)^{-1}(3)=c
$$
We also have $f: X \rightarrow Y$ such that
$$
f(1)=a, f(2)=b, f(3)=c \Rightarrow\left(f^{-1}\right)^{-1}=f \text {. }
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.