Inverse of a Function

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JEE Main 2008

Let $f : N \to Y$ be a function defined as $f (x) = 4 x + 3$ where $Y = {y \in N : y = 4 x + 3 for some x \in N}$ . Show that $f$ is invertible and its inverse is

g (y) = \frac{3 y + 4}{3}

(B)

g (y) = 4 + \frac{y + 3}{4}

(C)

g (y) = \frac{y + 3}{4}

(D)

g (y) = \frac{y - 3}{4}

JEE Advanced 2013

LEVELJEE Main

Let $f : (0, 1) \to R$ be defined by $f (x) = \frac{b - x}{1 - b x}$ , where $b$ is a constant such that $0 < b < 1$ . Then

* Multiple Correct Options

(A)

f

is not invertible on

(0, 1)

(B)

f \neq = f^{- 1}

(0, 1)

and

f^{'} (0) = 1/ f^{'} (0)

(C)

f = f^{- 1}

(0, 1)

and

f^{'} (0) = 1/ f^{'} (0)

(D)

f^{- 1}

is differentiable on

(0, 1)

JEE Advanced 2004

LEVELJEE Main

If $f (x) = sin x + cos x, g (x) = x^{2} - 1$ , then $g (f (x))$ is invertible in the domain

(A)

[0, π /2]

(B)

[- π /4, π /4]

(C)

[- π /2, π /2]

(D)

[0, π]

JEE Advanced 2002

LEVELJEE Main

Suppose $f (x) = (x + 1)^{2}$ for $x \geq - 1$ . If $g (x)$ is the function whose graph is the reflection of the graph of $f (x)$ with respect to the line $y = x$ , then $g (x)$ equals

(A)

- x - 1, x \geq 0

(B)

1/ (x + 1)^{2}, x > - 1

(C)

x + 1, x \geq - 1

(D)

x - 1, x \geq 0

JEE Advanced 2001

LEVELJEE Main

If $f : [1, \infty) \to [2, \infty)$ is given by $f (x) = x + 1/ x$ , then $f^{- 1} (x)$ equals

(A)

(x + x^{2} - 4) /2

(B)

x / (1 + x^{2})

(C)

(x - x^{2} - 4) /2

(D)

1 + x^{2} - 4

JEE Advanced 1999

LEVELJEE Main

If the function $f : [1, \infty) \to [1, \infty)$ is defined by $f (x) = 2^{x (x - 1)}$ , then $f^{- 1} (x)$ is

(A)

(1/2)^{x (x - 1)}

(B)

\frac{1}{2} (1 + 1 + 4 lo g_{2} x)

(C)

\frac{1}{2} (1 - 1 + 4 lo g_{2} x)

If $f (x) = 3 x - 5$ , then $f^{- 1} (x)$

(A)

is given by

\frac{1}{3 x - 5}

(B)

is given by

\frac{x + 5}{3}

(C)

does not exist because

f

is not one-one

(D)

does not exist because

f

is not onto

JEE Advanced 1995

LEVELJEE Main

Let $f (x) = (x + 1)^{2} - 1, x \geq - 1$ . Then the set ${x : f (x) = f^{- 1} (x)}$ is

(A)

{0, - 1, \frac{- 3 + i 3}{2}, \frac{- 3 - i 3}{2}}

(B)

{0, 1, - 1}

(C)

{0, - 1}

If $y = f (x) = \frac{x + 2}{x - 1}$ then

* Multiple Correct Options

(A)

x = f (y)

(B)

f (1) = 3

(C)

y

increases with

x

for

x < 1

(D)

f

is a rational function of

x

JEE Advanced 1982

LEVELJEE Main

Let $f$ be a one-one function with domain ${x, y, z}$ and range ${1, 2, 3}$ . It is given that exactly one of the following statements is true and the remaining two are false $f (x) = 1, f (y) \neq = 1, f (z) \neq = 2$ determine $f^{- 1} (1)$ .

JEE Advanced 1982

LEVELBoard

$y = 1 0^{x}$ is the reflection of $y = lo g_{10} x$ in the line whose equation is .........

JEE Main 2018 (15 April Evening)

LEVELJEE Main

Let $f : A \to B$ be a function defined as $f (x) = \frac{x - 1}{x - 2}$ , where $A = R - {2}$ and $B = R - {1}$ . Then f is :-

(A)

Invertible and

f^{- 1} (y) = \frac{2 y - 1}{y - 1}

f^{- 1} (y) = \frac{3 y - 1}{y - 1}

(D)

Invertible and

f^{- 1} (y) = \frac{2 y + 1}{y - 1}

JEE Main 2020 (8 January Shift 1)

LEVELJEE Main

The inverse function of $f (x) = \frac{8 ^{2 x} - 8 ^{- 2 x}}{8 ^{2 x} + 8 ^{- 2 x}}$ , $x \in (- 1, 1)$ , is

(A)

\frac{1}{4} (lo g_{8} e) lo g_{e} (\frac{1 - x}{1 + x})

(B)

\frac{1}{4} (lo g_{8} e) lo g_{e} (\frac{1 + x}{1 - x})

(C)

\frac{1}{4} lo g_{e} (\frac{1 + x}{1 - x})

(D)

\frac{1}{4} lo g_{e} (\frac{1 - x}{1 + x})

JEE Main 2020 - 8 Jan (Morning)

LEVELJEE Main

The inverse function of $f (x) = \frac{8 ^{2 x} - 8 ^{- 2 x}}{8 ^{2 x} + 8 ^{- 2 x}}$ , $x \in (- 1, 1)$ , is

(A)

\frac{1}{4} (lo g_{8} e) lo g_{e} (\frac{1 + x}{1 - x})

(B)

\frac{1}{2} lo g_{8} (\frac{1 - x}{1 + x})

(C)

\frac{1}{4} (lo g_{8} e) lo g_{e} (\frac{1 - x}{1 + x})

(D)

\frac{1}{2} lo g_{8} (\frac{1 + x}{1 - x})

JEE Main 2021 (25 July Shift 2)

LEVELJEE Main

Consider function $f : A \to B$ and $g : B \to C (A, B, C \subseteq R)$ such that $(g \circ f)^{- 1}$ exists then:

(A)

f

and

g

both are one-one

(B)

f

and

g

both are onto

(C)

f

is one-one and

g

is onto

(D)

f

is onto and

g

is one-one

JEE Main 2021 (18 March Shift 2)

LEVELJEE Main

Let $f : R - {3} \to R - {1}$ be defined by $f (x) = \frac{x - 2}{x - 3}$ . Let $g : R \to R$ be given as $g (x) = 2 x - 3$ . Then, the sum of all the values of $x$ for which $f^{- 1} (x) + g^{- 1} (x) = \frac{13}{2}$ is equal to

JEE Main 2023 (25 January Shift 1)

LEVELJEE Main

For some $a, b, c \in N$ , let $f (x) = a x - 3$ and $g (x) = x^{b} + c, x \in R$ . If $(f o g)^{- 1} (x) = (\frac{x - 7}{2})^{1/3}$ , then $(f o g) (a c) + (g o f) (b)$ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:N→Y be a function defined as f(x)=4x+3 where Y={y∈N:y=4x+3 for some x∈N}. Show that f is invertible and its inverse is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:(0,1)→R be defined by f(x)=1−bxb−x​, where b is a constant such that 0<b<1. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=sinx+cosx,g(x)=x2−1, then g(f(x)) is invertible in the domain

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Suppose f(x)=(x+1)2 for x≥−1. If g(x) is the function whose graph is the reflection of the graph of f(x) with respect to the line y=x, then g(x) equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f:[1,∞)→[2,∞) is given by f(x)=x+1/x, then f−1(x) equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the function f:[1,∞)→[1,∞) is defined by f(x)=2x(x−1), then f−1(x) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=3x−5, then f−1(x)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=(x+1)2−1,x≥−1. Then the set {x:f(x)=f−1(x)} is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If y=f(x)=x−1x+2​ then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y=10x is the reflection of y=log10​x in the line whose equation is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:A→B be a function defined as f(x)=x−2x−1​, where A=R−{2} and B=R−{1}. Then f is :-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The inverse function of f(x)=82x+8−2x82x−8−2x​, x∈(−1,1), is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The inverse function of f(x)=82x+8−2x82x−8−2x​, x∈(−1,1), is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider function f:A→B and g:B→C(A,B,C⊆R) such that (g∘f)−1 exists then:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For some a,b,c∈N, let f(x)=ax−3 and g(x)=xb+c,x∈R. If (fog)−1(x)=(2x−7​)1/3, then (fog)(ac)+(gof)(b) is equal to

Let $f : N \to Y$ be a function defined as $f (x) = 4 x + 3$ where $Y = {y \in N : y = 4 x + 3 for some x \in N}$ . Show that $f$ is invertible and its inverse is

Let $f : (0, 1) \to R$ be defined by $f (x) = \frac{b - x}{1 - b x}$ , where $b$ is a constant such that $0 < b < 1$ . Then

If $f (x) = sin x + cos x, g (x) = x^{2} - 1$ , then $g (f (x))$ is invertible in the domain

Suppose $f (x) = (x + 1)^{2}$ for $x \geq - 1$ . If $g (x)$ is the function whose graph is the reflection of the graph of $f (x)$ with respect to the line $y = x$ , then $g (x)$ equals

If $f : [1, \infty) \to [2, \infty)$ is given by $f (x) = x + 1/ x$ , then $f^{- 1} (x)$ equals

If the function $f : [1, \infty) \to [1, \infty)$ is defined by $f (x) = 2^{x (x - 1)}$ , then $f^{- 1} (x)$ is

If $f (x) = 3 x - 5$ , then $f^{- 1} (x)$

Let $f (x) = (x + 1)^{2} - 1, x \geq - 1$ . Then the set ${x : f (x) = f^{- 1} (x)}$ is

If $y = f (x) = \frac{x + 2}{x - 1}$ then

$y = 1 0^{x}$ is the reflection of $y = lo g_{10} x$ in the line whose equation is .........

Let $f : A \to B$ be a function defined as $f (x) = \frac{x - 1}{x - 2}$ , where $A = R - {2}$ and $B = R - {1}$ . Then f is :-

The inverse function of $f (x) = \frac{8 ^{2 x} - 8 ^{- 2 x}}{8 ^{2 x} + 8 ^{- 2 x}}$ , $x \in (- 1, 1)$ , is

The inverse function of $f (x) = \frac{8 ^{2 x} - 8 ^{- 2 x}}{8 ^{2 x} + 8 ^{- 2 x}}$ , $x \in (- 1, 1)$ , is

Consider function $f : A \to B$ and $g : B \to C (A, B, C \subseteq R)$ such that $(g \circ f)^{- 1}$ exists then:

For some $a, b, c \in N$ , let $f (x) = a x - 3$ and $g (x) = x^{b} + c, x \in R$ . If $(f o g)^{- 1} (x) = (\frac{x - 7}{2})^{1/3}$ , then $(f o g) (a c) + (g o f) (b)$ is equal to