Inverse Function: Definition and Examples

An inverse function is the opposite of a function. A function receives an input value and gives back an output while the inverse functions gives the input value with the help of the output. In simple terms, if a function $f$ maps $x$ to $y$, the inverse function, denoted $f^{-1}$. maps $y$ back to $x$. For example, an exponential function can be undone using a logarithm. In this case, logarithm is the inverse of an exponential function.

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.

Inverse 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 $B$.

A function accepts values, performs particular or specific operations on these values and generates an output or a result. The inverse function uses the output and then operates and reaches back to the original function as in the beginning.

What is the Inverse Function?

The inverse function is the reverse of the function. The inverse function returns the original value for which a function gave the output. The inverse of a function $f$ is denoted by $f^{-1}$ and it exists only when the condition that $f$ is both one-one and onto function is satisfied.

The composition of the function $f$ and the reciprocal function $f^{-1}$ returns us back the domain value of $x$.

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

For a function ' $f$ ' to be considered an inverse function, each element in the range $y$ $\in Y$ has been mapped from some element $x \in X$ in the domain set, and such a relation is called a one-one relation or an injection relation. Also the inverse $f^{-1}$ of the given function has a domain $y \in Y$ is related to a distinct element $x \in X$ in the codomain set, and this kind of relationship with reference to the given function ' $f$ ' is an onto function or a surjection function. Thus, when both the criteria are met, the function is called a bijective function.

Types of inverse function

The types of inverse functions are,

Inverse Trigonometric Functions

They are also known as arc functions because of the reason that they produce the length of the arc, which is required to obtain that particular value. There are six inverse trigonometric functions which include arcsine $\left(\sin ^{-1}\right)$, arccosine $\left(\cos ^{-1}\right)$, arctangent $\left(\tan ^{-1}\right)$, arcsecant $\left(\mathrm{sec}^{-1}\right)$, arccosecant $\left(\operatorname{cosec}^{-1}\right)$, and arccotangent $\left(\mathrm{cot}^{-1}\right)$.

Inverse Rational Function

A rational function is a function of form $f(x)=P(x) / Q(x)$ where $Q(x) \neq 0$. To find the inverse of a rational function, we follow the following steps.
- Step 1: We replace $f(x)=y$.
- Step 2:Then we interchange $x$ and $y$.
- Step 3: Fuethur, we solve for y in terms of x.
- Step 4: Finally, we 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 $\sinh ^{-1}, \cosh ^{-1}, \tanh ^{-1}, \operatorname{csch}^{-1}, \operatorname{coth}^{-1}$, and $\operatorname{sech}^{-1}$.

Inverse Logarithmic Functions

In simple words, the inverse log function is the process that cancels out a logarithmic function's effect. Or we can say that it undoes the effect of log function.

For example, if we 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.Hence, we can see that the effect of log has been eliminated from the equation.

Inverse Function Formulas

Inverse function formulas include the method on how to find inverse functions and how to check inverse functions.

How to find Inverse Functions?

The steps to find the inverse functions are,

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 $x$.

How to Check Inverse Functions?

To check whether the given function is the inverse of another function, let us calculate the composite of those two functions. If the composition of two functions results as the input value, then the given function is the inverse function.

Let $f$ and $g$ be two functions. For these functins $f$ and $g$ be inverse of each other, the composition of $f$ and $g$, $f \circ g (x) = f(g(x)) = g(f(x)) = x$.

Inverse Function Graph

The graph of the inverse function is similar to the graph of the original function. The only difference is the exchange in the $x$ and $y$ coordinates of the graph. That is, the values of $x$ in the graph of the original function will be the values of $y$ in the graph of the inverse function and vice versa. For instance, a point ( $x, y$ ) in the original function will have a corresponding point $(y, x)$ in the inverse function.

In other words, the inverse function is the reflection of the original function across the line $y=x$.

Inverse Function Graph Examples

Let us look into some of the inverse function graph examples.

1. The inverse of function of $\sin x$ is $\sin ^{-1} x$.

Graph of $\sin x$:

Graph of $\sin ^{-1} x$:

Comparison of graph of $\sin x$ and $\sin ^{-1} x$:

2. The function $e^x$ is the inverse of $\ln x$.

Comparison of graph of $e^x$ and $\ln x$:

3. The inverse function of $y=x^2$ is $\sqrt(y)$

Graph of $y=x^2$:

Graph of $y=x^2$:

Properties of an inverse function

The properties of the inverse functions are,

i) The inverse of a bijection is always unique.

ii) if $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ is a bijection and $\mathrm{g}: \mathrm{B} \rightarrow \mathrm{A}$ 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 $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ and $\mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}$ 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 $y = x$.

Recommended Video Based on Inverse of Functions

Solved Examples Based On 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

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

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) $y=\operatorname{cosec} x$
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 $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \sin ^{-1} y=-1 \leq x \leq 1$. 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) $x^2+4 x+1$
3) $x^2+4 x+4$
4) $x^2+x+1$

Solution:

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

Squaring both sides

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

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

Hence, the answer is the option 2.

List of Topics Related to Inverse Functions