Composition of function: Conditions and Properties

An inverse function or an anti-function is defined as a function, which can reverse into another function. The inverse of a function reverses the operation of the function, mapping outputs back to their corresponding inputs. If a function f maps x to y, the inverse function, denoted $f^{-1}$. maps y back to x.

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In this article, we will cover the concepts of the inverse of a function. This concept falls under the broader category of relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of five questions have been asked on this concept, including one in 2018, one in 2020, and three in 2021.

Function-

A relation from a set $A$ to a set $B$ is said to be a function from $A$ to $B$ if every element of set $A$ has one and only one image in set $B$.

OR
$A$ and $B$ are two non-empty sets, then a relation from $A$ to $B$ is said to be a function if each element $x$ in $A$ is assigned a unique element $f(x)$ in $B$, and it is written as
f: $A\to B$ and read as $f$ is mapping from $A$ to $B$.
The third one is not a function because $d$ is not related(mapped) to any element in B.
Fourth is not a function as element a in A is mapped to more than one element in B .
Inverse of a function
Function $\mathrm{f}:\mathrm{X}\to \mathrm{Y}$ is an invertible function if it is one-one and onto
Also, its inverse g is defined in the following way
$g:Y\to X$ 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, $\mathrm{A}=$ $\{1,2,3,4\}$ and $B=\{2,4,6,8\}$ and $f:A\to B$ is given $f(x)=2x$, then write $f$ and $f^{-1}$ as a set of ordered pairs.

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

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

Types of inverse function

Inverse Trigonometric Functions

The inverse trigonometric functions are also known as arc functions as they produce the length of the arc, which is required to obtain that particular value. There are six inverse trigonometric functions which include arcsine (sin^-1), arccosine (cos^-1), arctangent (tan^-1), arcsecant (sec^-1), arccosecant (cosec^-1), and arccotangent (cot^-1).

Inverse Rational Function

A rational function is a function of form $f(x)=P(x)/Q(x)$ where $Q(x)\ne 0$. To find the inverse of a rational function, follow the following steps. An example is also given below which can help you to understand the concept better.
- Step 1: Replace $f(x)=y$
- Step 2: Interchange $x$ and $y$
- Step 3: Solve for y in terms of x
- Step 4: Replace $y$ with ${\mathrm{f}}^{-1}(\mathrm{x})$ and the inverse of the function is obtained.

Inverse Hyperbolic Functions
Just like inverse trigonometric functions, the inverse hyperbolic functions are the inverses of the hyperbolic functions. There are mainly 6 inverse hyperbolic functions that exist which include ${\sinh }^{-1},{\cosh }^{-1},{\tanh }^{-1},{\mathrm{csch}}^{-1},{\coth }^{-1}$, and ${\mathrm{sech}}^{-1}$.

Inverse Logarithmic Functions

In essence, the inverse log function is the process that cancels out a logarithmic function's effect.
The idea of inverses frequently piques my interest the most when working with mathematical functions because it enables me to reverse the effects of a function and return to the initial value prior to the function's application.

For example, if I have a function $f(x)={\log }_b{x}_{\text{its inverse function would be written as}}{f}^{-1}(y)=b^y$, essentially using the base of the logarithm to re-exponentiate the value.

Realising that the inverse log function is merely the exponential function with the same base as the logarithm is essential to comprehending its characteristics and behaviour.

A key idea in mathematics is the reciprocal link between the logarithmic and exponential functions. I believe that investigating this link provides a better understanding of the symmetry present in mathematics and goes beyond merely calculating equations.

Determining the inverse function is a graceful example of how entangled operations can break down complicated expressions into their simpler parts. Come along with me as we explore this fascinating mathematical journey; you won't regret it!

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 $\mathrm{f}:\mathrm{A}\to \mathrm{B}$ is a bijection and $\mathrm{g}:\mathrm{B}\to \mathrm{A}$ is the inverse of f , then $f\circ g=I_B$ and $g\circ f=I_A$, where ${\mathrm{I}}_{\mathrm{A}}$ and ${\mathrm{I}}_{\mathrm{B}}$ are identity functions on the sets A and B , respectively.
iii) The inverse of a bijection is also a bijection.
iv) If $\mathrm{f}:\mathrm{A}\to \mathrm{B}$ and $\mathrm{g}:\mathrm{B}\to \mathrm{C}$ 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.

Summary

In general, an inverse function is one that "undoes" the operation of another function. An exponential function can be undone using a logarithm. Exponential function equations are solved with logarithms. Mathematicians utilise exponential functions to solve real-world issues requiring exponential growth and decay, including those involving radioactivity or population dynamics. High school algebra classes usually cover exponential functions and logarithms.

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Solved Examples Based On the Inverse of Functions:

Example 1: What is the inverse of $x^5$ ?
1) $\sqrt{x}$
2) $x^{\frac{1}{5}}$
3) $x^5$
4) $x$

Solution:
Property of Inverse Function -
The inverse of a bijection is unique.
Since ${\left(x^5\right)}^{\frac{1}{5}}=x$, the inverse of $x^5$ is $x^{\frac{1}{5}}$
Hence, the answer is the option 2.
Example 2: What is the inverse of $f(x)=x^{\frac{1}{3}}+1$ ?
1) $x^3-1$
2) $x^{\frac{1}{3}}-1$
3) $(x-1{)}^3$
4) $(x-1{)}^{\frac{1}{3}}$

Solution:

4) None of these

$\begin{array}{rl}& \text{If}g(x)=(x-1{)}^3\\ & g\circ f(x)=g(x^{\frac{1}{3}}+1)={(x^{\frac{1}{3}}+1-1)}^3=x\end{array}$

Hence, the answer is the option 3.
Example 3: The inverse of the function $y=\sqrt{x-2}$
1) $x^2+1$
2) $x^2+2$
3) $x^2+3$
4) $x^2-2$

Solution:
$g(x)=\sqrt{x-2}\Rightarrow g^2(x)=x-2\Rightarrow x=g^2(x)+2\Rightarrow y=x^2+2$ is the inverse.
Hence, the answer is the option 2.
Example 4: The inverse of the function $y=\sin x$ is
1) $y=\mathrm{cosec}x$
2) $y={\sin }^{-1}x$
3) $y=\frac{1}{\sin x}$

Solution:

Property of Inverse -

The inverse of a bijection is also a bijection.

Inverse of $y=\sin x$, where $-\frac{\pi }{2}\le x\le \frac{\pi }{2}{\sin }^{-1}y=-1\le x\le 1$. We need to specify the value of x so that it is a bijective function
Hence, the answer is the option 2.
Example 5 : What is the inverse of $y=\sqrt{x+3}-2$ ?
1) $x^2+x+4$
2) $x^2+4x+1$
3) $x^2+4x+4$
4) $x^2+x+1$

Solution:

$y=\sqrt{x+3}-2\Rightarrow y+2=\sqrt{x+3}$

Squaring both sides

$\begin{array}{rl}& y^2+4y+4=x+3\\ & x=y^2+4y+1\end{array}$

Thus inverse is $y=x^2+4x+1$

Hence, the answer is the option 2.

Frequently Asked Questions(FAQ)-

1. What is a function?

Ans: Functions are one of the basic concepts in mathematics that have numerous applications in the real world.

2. What is the inverse function?

Ans: An inverse function or an anti-function is defined as a function, which can reverse into another function.

3. Is the inverse of a bijection unique?

Ans: Yes, the inverse of bijection is unique.

4. If $f(x)=x^{1/3}$; then the function $g(x)=x^3$ is the inverse of $\mathrm{f}(x)$ for

Ans: Property of Inverse
The inverse of a bijection is also a bijection.
Both $y=x^3.$. and.. $y=x^{1/3}$ are bijective function

5. Write some types of inverse functions.

Ans: Inverse trigonometric function, inverse rational function, inverse hyperbolic function, etc are some types of inverse functions.