Inverse of a function - Practice Questions & MCQ

Function $\mathrm{f}: \mathrm{X} \rightarrow \mathrm{Y}$ is an invertible function if it is one-one and onto
Also, its inverse g is defined in the following way
$g: Y \rightarrow X$ such that if $f(a)=b$, then $g(b)=a$
The function $g$ is called the inverse of $f$ and is denoted by $f^{-1}$.
Let us consider a one-one and onto function $f$ with domain $A$ and co-domain $B$. Where, $A=\{1,2,3,4\}$ and $B=\{2,4,6,8\}$ and $f: A \rightarrow B$ is given $f(x)=2 x$, then write $f$ and $f^{-1}$ as a set of ordered pairs.

So, $f=\{(1,2)(2,4)(3,6)(4,8)\}$
And $\mathrm{f}^{-1}=\{(2,1)(4,2)(6,3)(8,4)\}$

In above definition domain of $f=\{1,2,3,4\}=$ range of $f-1$
Range of $f=\{2,4,6,8\}=$ domain of $f^{-1}$.

Steps to find the inverse of a function:

i) First we write $f(x)$ as $y$ and equate $y=f(x)$, where $f(x)$ is a function in $x$
ii) Then we separate the variable $x$ as the dependent variable and express it in terms of $y$ by assuming $y$ as the independent variable
iii) Then we write $g(\mathrm{y})=\mathrm{x}$ where $\mathrm{g}(\mathrm{y})$ is a function in y
iv) And finally, we replace every $y$ by $x$

Properties of an inverse function

i) The inverse of a bijection is unique. 

ii) if f∶ A → B is a bijection and g∶ B → A is the inverse of f, then $f o g=I_B$ and $g \circ f=I_A$ , where I_A   and I_B   are identity functions on the sets A and B, respectively.

iii) The inverse of a bijection is also a bijection.

iv) If f: A → B and g: B → C  are two bijections, then $(\text { got })^{-1}=\mathrm{f}^{-1} \mathrm{og}^{-1}$  

v) The graphs of f and its inverse function, are mirror images of each other in the line y = x.