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Inverse of a function - (Concept)
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 IA and IB 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.
| Exam | Chapter |
| JEE MAIN | Sets, Relations and Functions |
What is the inverse of ?
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What is inverse of ?
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The inverse of the function
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The inverse of the function is
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What is the inverse of
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If then what is the inverse of function f(x)?
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If ; then the function
is the inverse of f(x) for
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If then find g(x) such that it is the inverse of f(x)
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If
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The inverse function of $f(x)=\frac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}},x\epsilon (-1,1),$ is:
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Let be defined by
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Let be given as
. Then , the sum of all values of x for which
is equal to:
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If . Find the inverse of f(x)
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Which of the following functions from f: A->A is invertible, where x [-1,1]?
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If . Then
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Which of the following functions is inverse of itself in its domain?
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Consider function $f: \mathrm{A} \rightarrow \mathrm{B}$ and $g: B \rightarrow C(A, B, C \subseteq R)$ such that (gof) ${ }^{-1}$ exists, then
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Let be a function defined as
, where
and
. Then
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The inverse of is :
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$f(x)$ and $g(x)$ are inverse of bijective function $h(x)$ then:
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Find the domain of the function $f(x)=\frac{2}{\log _{10}(1-x)}+\sqrt{x+3}$
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If $f(x)=x^3$ and $g(x)=\tan x$, where $f: R \rightarrow R$ and $g:\left(\frac{-\pi}{2}, \frac{\pi}{2}\right) \rightarrow R$, then $(f \circ g)^{-1}(x)=$
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Inverse of a bijective function is :
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Let $\mathrm{f}(\mathrm{x})=x^3+4$ is bijective, then its inverse is
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The inverse of the function $y=\left [ 1-\left ( x-3 \right )^{4} \right ]^{\frac{1}{7}}$ is
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If $f(x)=x^2+2 x, x \geq 1$, then $f^{-1}(x)$ equals
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Let $f:[0,1] \rightarrow[0,1]$ be defined by $f(x)=\{x$, if is rational $1-x$, if is irrational\}.
Then (fof) $x$ is
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If $f: R \rightarrow R$ be given by $f(x)=\tan x$, then $f^{-1}(1) i s$
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If $f(x) = \frac{4x + 3}{6x - 4} , x \neq \frac{2}{3}$, then show that (fof) (x) = x, for all $x \neq \frac{2}{3}$ . Also, write inverse of f.
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Let $f:\{R-3 / 5\} \rightarrow R$ be defined by $f(x)=(3 x+2) /(5 x-3)$ then
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Let $\mathrm{f}: R \rightarrow R$ be defined by
$
f(x)= \begin{cases}2 x & \text { when } x>3 \\ x^2 & \text { when } 1<x \leq 3 \\ 3 x & \text { when } x \leq 1\end{cases}
$
The expression $(f(-1)+f(2)+f(4))$ is :
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$f(x)=x^2-2 a x+a(a+1), f:[a, \infty) \rightarrow[a, \infty)$. If one of the solution of the equation $f(x)=f^{-1}(x)$ is 100, then other solution may be:
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If $f(x)=\sqrt[4]{\frac{17}{\log _5(10-25 x)}-1}$, the value of $a^{\prime} a^{\prime}$ for which satisfies $f^{-1}(2 a-4)=\frac{1}{5}$ is:
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If $f(x)$ is an invertible function and $g(x)=2 f(x)+9$ then, $g^{-1}(x)$ is
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Find the inverse of $f: R \rightarrow R$ where $f(x)=x^2+2$.
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Definition 9 A function f : X Y is defined to be invertible, if there exists a function
g : Y X such that gof = I
and fog = I
. The function g is called the inverse of f and
is denoted by .
