Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced

Problem 5

If $f(x)=\frac{2 x}{1+x^{2}}$, show that, $f(\tan \theta)=\sin 2 \theta$

Varsha Aggarwal

Numerade Educator

01:05

Problem 6

If $f(x)=\log \left(\frac{x}{x-1}\right)$, show that,
$f(x+1)+f(x)=\log \left(\frac{x+1}{x-1}\right)$

Varsha Aggarwal

Numerade Educator

Problem 7

If $f(x)=\log \left(\frac{1+x}{1-x}\right)$, show that, $f\left(\frac{2 x}{1+x^{2}}\right)=2 f(x)$

Varsha Aggarwal

Numerade Educator

02:36

Problem 8

If $f(x)=\cos \left(\log _{e} x\right)$, find the value of
$f(x) f(y)-\frac{1}{2}(f(x)+f(y))$

Varsha Aggarwal

Numerade Educator

00:39

Problem 9

If $y=f(x)=\frac{a x-b}{b x-a}$, show that $x=f(y)$

Varsha Aggarwal

Numerade Educator

Problem 10

If for non zero $x, a f(x)+b f\left(\frac{1}{x}\right)=\frac{1}{x}-5$,
where $a \neq b$, find $f(x)$.

Varsha Aggarwal

Numerade Educator

Problem 11

If $f(x)=\frac{9^{x}}{9^{x}+3}$, then prove that $f(x)+f(1-x)=1$.

Varsha Aggarwal

Numerade Educator

02:10

Problem 11

$f(x)=\frac{9^{x}}{9^{x}+3}$, then prove that $f(x)+f(1-x)=1$.

Sahil Kumar

Numerade Educator

01:49

Problem 12

If $f\left(\frac{2 \tan x}{1+\tan ^{2} x}\right)=\frac{1}{2}(1+\cos 2 x)\left(\sec ^{2} x+2 \tan x\right)$
then find $f(x)$.

Varsha Aggarwal

Numerade Educator

12:40

Which one of the followings are functions? If so then find its Domains and Ranges. If not, then find when it can be a function and also find their Domains and Ranges..
(i) $y=m x+c$
(ii) $y=0$
(iii) $x=0$
(iv) $y=5$
(v) $x=3$
(vi) $x^{2}+y^{2}=9$
(vii) $y^{2}=4 x$
(viii) $x^{2}=4 y$
(ix) $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$
(x) $\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$
(xi) $x^{2}-y^{2}=4$
(xii) $y=(x-1)(x-2)(x-3)$

William Semus

Numerade Educator

Problem 14

in of Real Functions Find the domain of each of the following functions.
(i) $f(x)=\frac{x-2}{x+3}$
(ii) $f(x)=\frac{x+4}{x-3}$
(iii) $f(x)=\frac{1}{x}-1$
(iv) $f(x)=\frac{1}{x}+1$
(v) $f(x)=\frac{x-1}{x^{2}-3 x+2}$
(vi) $f(x)=\frac{x^{2}-5 x+4}{x^{2}-3 x+2}$
(vii) $f(x)=\frac{x^{2}+4}{x^{2}+2}$
viii) $f(x)=\frac{x^{2}+9}{x^{2}-4}$
(ix) $f(x)=\frac{x^{2}+x+1}{x^{2}-x+1}$\text { (x) } f(x)=\frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7}

Carson Merrill

Numerade Educator

Problem 15

Find the domain of each of the following functions
(i) $f(x)=\sqrt{x-2}$
(ii) $f(x)=\sqrt{x+5}$
(iii) $f(x)=\sqrt{4-x}$
(iv) $f(x)=\sqrt{x-2}+\sqrt{4-x}$
(v) $f(x)=\sqrt{x-3}-\sqrt{7-x}$
(vi) $f(x)=\sqrt{\frac{x-2}{5-x}}$
(vii) $f(x)=\sqrt{\frac{x}{x-1}}$
(viii) $f(x)=\sqrt{1-\frac{1}{x}}$
(ix) $f(x)=\sqrt{\frac{1}{x}-1}$
(x) $f(x)=\sqrt{\frac{x-2}{x}}$

Carson Merrill

Numerade Educator

Problem 16

Find the domain of each of the following functions
(i) $f(x)=\sqrt{|x|-2}$
(ii) $f(x)=\sqrt{4-|x|}$
(iii) $f(x)=\sqrt{|x|-1}+\sqrt{4-\lfloor x \mid}$
(iv) $f(x)=\sqrt{\frac{|x|-1}{3-|x|}}$
(v) $f(x)=\sqrt{|x|-x}$
(vi) $f(x)=\frac{1}{\sqrt{|x|-x}}$
(vii) $f(x)=\frac{1}{\sqrt{x-|x|}}$
viii) $f(x)=\sqrt{[x]-x}$
(ix) $f(x)=\frac{1}{\sqrt{[x]-x}}$
(x) $f(x)=\frac{1}{\sqrt{x-[x]}}$

Carson Merrill

Numerade Educator

00:49

Problem 17

$$
\text { Find the domain of } f(x)=\frac{1}{[x-2]}
$$

Varsha Aggarwal

Numerade Educator

Problem 18

$$
\text { Find the domain of } f(x)=\frac{x+3}{[x+1]}
$$

Susan Cooper

Numerade Educator

00:48

Problem 19

$$
\text { Find the domain of } f(x)=\frac{x^{2}+x+1}{[x]-4}
$$

Varsha Aggarwal

Numerade Educator

01:41

Problem 20

$$
\text { Find the domain of } f(x)=\frac{1}{\sqrt{x-[x]}}
$$

Varsha Aggarwal

Numerade Educator

Problem 21

$$
\text { Find the domain of } f(x)=\frac{1}{\sqrt{[x]-x}}
$$

Varsha Aggarwal

Numerade Educator

00:53

Problem 22

$$
\begin{aligned}
&\text { Let } f(x)=\cos \left[\pi^{2}\right] x+\cos \left[-\pi^{2}\right] x \\
&\text { Then find } f\left(\frac{\pi}{2}\right), f(\pi)
\end{aligned}
$$

Varsha Aggarwal

Numerade Educator

00:43

Problem 23

$$
\begin{aligned}
&\text { Find the exact value of }\\
&\left[\log _{e} 1\right]+\left[\log _{e} 2\right]+\left[\log _{e} 3\right]+\ldots+\left[\log _{e} 66\right]
\end{aligned}
$$

Jeyasree R T

Numerade Educator

Problem 24

$$
\text { Solve for }[x]^{2}-3[x]+2=0
$$

Varsha Aggarwal

Numerade Educator

Problem 25

$$
\text { Solve for }[x]^{2}-2[x]-8=0
$$

Varsha Aggarwal

Numerade Educator

03:03

Problem 26

$$
\text { Find the number of solution of } x^{2}-4-[x]=0
$$

Varsha Aggarwal

Numerade Educator

00:51

Problem 27

Find the value of
$\left[\frac{1}{4}\right]+\left[\frac{1}{4}+\frac{1}{100}\right]+\left[\frac{1}{4}+\frac{2}{100}\right]+\ldots+\left[\frac{1}{4}+\frac{99}{100}\right]$
where $[.]=$ G.I.F

Varsha Aggarwal

Numerade Educator

04:09

Problem 28

$$
\begin{aligned}
&\text { Eind the value of }\\
&[1007]+\left[503 \frac{3}{4}\right]+\left[252 \frac{1}{8}\right]+\left[126 \frac{5}{16}\right]+\ldots
\end{aligned}
$$

Anas Venkitta

Numerade Educator

Problem 29

$$
\text { Solve the equation }[\sin x]=0
$$

Varsha Aggarwal

Numerade Educator

01:17

Problem 30

If $y=2[x]+3=3[x-2]+5$, then find the value of $[x+y]$

Varsha Aggarwal

Numerade Educator

Problem 31

Find the doamin of $f(x)=\sqrt{[x]-2}$

Varsha Aggarwal

Numerade Educator

Problem 32

Find the domain of $f(x)=\sqrt{4-[x]}$

Varsha Aggarwal

Numerade Educator

01:35

Problem 33

Find the domain of $f(x)=\sqrt{[x]-1}+\sqrt{3-[x]}$

Varsha Aggarwal

Numerade Educator

02:46

Problem 34

Find the domain of $f(x)=\sqrt{\frac{[x]-2}{5-[x]}}$

Sahil Kumar

Numerade Educator

04:03

Problem 35

Find the value of
(i) $\left[\frac{\sin ^{2} x}{2}\right]$
(ii) $\left[\sin ^{2}\left(\frac{\pi}{2}\right) x\right]$
(iii) $\left[\cos ^{2}\left(\frac{\pi}{2}\right) x\right]$
(iv) $\left[\frac{\cos ^{2} x}{2014}\right]$.

Sahil Kumar

Numerade Educator

01:26

Problem 36

Find the set of values of $x$ satisfying $[\sin x]+[\cos x]=1, \forall x \in[0,2 \pi]$

Varsha Aggarwal

Numerade Educator

01:43

Problem 37

If $x$ satisfies the equation $[x+0.19]+[x+0.20]+[x+0.21]+\ldots+[x+0.22]$
$+\ldots+[x+0.91]=542$
Then find the value of $[100 x]$

Varsha Aggarwal

Numerade Educator

01:27

Problem 38

Find the value of
$\left[\frac{2015}{2}\right]+\left[\frac{2016}{4}\right]+\left[\frac{2018}{8}\right]+\left[\frac{2022}{16}\right]+\ldots$

Varsha Aggarwal

Numerade Educator

02:27

Problem 39

Prove that $[2 x]=[x]+\left[x+\frac{1}{2}\right]$

Varsha Aggarwal

Numerade Educator

02:37

Problem 40

Prove that $[3 x]=[x]+\left[x+\frac{1}{3}\right]+\left[x+\frac{2}{3}\right]$

Varsha Aggarwal

Numerade Educator

01:05

Problem 41

Find the number of integral values of $\left[\frac{2 x^{2}}{x^{2}+1}\right]$

Varsha Aggarwal

Numerade Educator

00:42

Problem 42

Solve for $x:\{x\}+\{\sin x\}=2$

Varsha Aggarwal

Numerade Educator

01:37

Problem 43

If $\{x\}$ and $[x]$ represent fractional and integral part of $x$ respectively, then find the value of
$[x]+\sum_{r=1}^{2014}\left(\frac{\{x+r\}}{2014}\right)$.

Varsha Aggarwal

Numerade Educator

02:00

Problem 44

Solve for $x: 4\{x\}=x+[x]$, where $\{x\}$ and $[x]$ are the fractional and integral part of $x$ respectively.

Varsha Aggarwal

Numerade Educator

02:00

Problem 45

Solve for $x: 2 x+3\{x\}=4[x]-2$

Varsha Aggarwal

Numerade Educator

01:51

Problem 46

Solve for $x: x^{2}-4 x+[x]+3=0$

Varsha Aggarwal

Numerade Educator

Problem 47

Find the number of values of $x$ satisfying $\left\{x^{2}\right\}+\left[x^{4}\right]=1$

Varsha Aggarwal

Numerade Educator

01:27

Problem 48

Find the number of solutions of
$(x\}+\{\tan \pi x)=0$

Varsha Aggarwal

Numerade Educator

01:07

Problem 49

Find the number of integral values of $y=\left\{\frac{x^{2}+1}{2}\right\}, 0 \leq x \leq 2$

Varsha Aggarwal

Numerade Educator

02:03

Problem 50

Find the values of $x$ satisfying the equation $\left\{\frac{\cos ^{2} x-2}{2}\right\}=\frac{1}{4}$

Varsha Aggarwal

Numerade Educator

01:21

Problem 51

Find the domain of the function
$f(x)=\sqrt{2\{x\}^{2}-3\{x\}+1}, x>0$

Varsha Aggarwal

Numerade Educator

Problem 52

Find the max and min values of
$f(x)=3 \sin x+4 \cos x+10$

Varsha Aggarwal

Numerade Educator

02:02

Problem 52

Find the $\max$ and min values of
$f(x)=3 \sin x+4 \cos x+10$

Sahil Kumar

Numerade Educator

02:16

Problem 53

Find the range of $f(x)=\sin x+\cos x+3$

Sahil Kumar

Numerade Educator

01:15

Problem 54

Let $A=\sin ^{4} \theta+\cos ^{4} \theta$, then find $A$

Varsha Aggarwal

Numerade Educator

01:33

Problem 55

Find the max and min values of
$$
f(\theta)=\sin ^{6} \theta+\cos ^{6} \theta
$$

Varsha Aggarwal

Numerade Educator

03:09

Problem 56

If $A=\cos ^{2} \theta+\sin ^{4} \theta$ and $B=\cos ^{4} \theta+\sin ^{2} \theta$ such that
$m_{1}=\operatorname{Max}$ of $A$ and $m_{2}=$ Min of $B$ then find the value of $m_{1}^{2}+m_{2}^{2}+m_{1} m_{2}$

Varsha Aggarwal

Numerade Educator

02:16

Problem 57

Find the max and min values of
$f(\theta)=\sin ^{2}(\sin \theta)+\cos ^{2}(\cos \theta)$

Varsha Aggarwal

Numerade Educator

01:23

Problem 58

Find the minimum value of
$f(\theta)=(3 \sin (\theta)-4 \cos (\theta)-10)$
$(3 \sin (\theta)+4 \cos (\theta)-10)$

Varsha Aggarwal

Numerade Educator

01:34

Problem 59

Find the range of $A=\sin ^{2010} \theta+\cos ^{2014} \theta$

Varsha Aggarwal

Numerade Educator

Problem 60

Find the minimum value of $f(x)=\frac{x^{2} \sin ^{2} x+4}{x \sin x}$, where
$x \in\left(0, \frac{\pi}{2}\right)$

Varsha Aggarwal

Numerade Educator

Problem 61

Find the domain of $f(x)=e^{\frac{x}{2}-1}$

Varsha Aggarwal

Numerade Educator

Problem 62

Find the domain of $f(x)=\left|e^{x}-2\right|$

Varsha Aggarwal

Numerade Educator

01:16

Problem 63

Find the domain of $f(x)=\left|e^{\operatorname{lt} \mid}-2\right|$

Varsha Aggarwal

Numerade Educator

Problem 64

Find the domain of $f(x)=\left|e^{-\operatorname{lx} \mid}-\frac{1}{2}\right|$

Varsha Aggarwal

Numerade Educator

02:01

Problem 65

Find the domain of $f(x)=\sqrt{e^{-\mathrm{td}}-\frac{1}{2}}$

Sahil Kumar

Numerade Educator

Problem 66

Find the domain of $f(x)=\left(e^{2 x}+e^{x}+1\right)$

Varsha Aggarwal

Numerade Educator

Problem 67

Find the domain of $f(x)=\left(e^{x}+1\right)^{2}+3$

Varsha Aggarwal

Numerade Educator

Problem 68

Find the domain of $f(x)$ satisfying $e^{x}+e^{f(x)}=e$

Varsha Aggarwal

Numerade Educator

Problem 69

Find the number of solutions of $2^{x}+3^{x}+4^{x}-5^{x}=0 .$

Varsha Aggarwal

Numerade Educator

Problem 70

Find the number of solutions of $1+3^{x / 2}=2^{x}$

Varsha Aggarwal

Numerade Educator

01:09

Problem 71

Find the domain of the function
$f(x)=\log _{2}\left(x^{2}-4 x+3\right)$

Varsha Aggarwal

Numerade Educator

Problem 72

Find the domain of the function
$f(x)=e^{\frac{1}{2}-1}+\log (1-x)+x^{1001}$

Varsha Aggarwal

Numerade Educator

00:47

Problem 73

Find the domain of the function
$f(x)=\log \left(\frac{x}{x-2}\right)$

Varsha Aggarwal

Numerade Educator

00:48

Problem 74

Find the domain of the function
$f(x)=\frac{1}{\log (x-2)}$

Varsha Aggarwal

Numerade Educator

01:09

Problem 75

Find the domain of the functions
$f(x)=\frac{\log (1-x)}{x^{2}-3 x+2}$

Varsha Aggarwal

Numerade Educator

Problem 76

Find the domain of the function
$f(x)=\frac{\log (3-x)}{x^{2}-5 x+4}$

Varsha Aggarwal

Numerade Educator

01:39

Problem 77

Find the domain of the function
$f(x)=\frac{1}{\log \left(9-x^{2}\right)}$

Varsha Aggarwal

Numerade Educator

00:50

Problem 78

Find the domain of the function
$f(x)=\log _{1 / 2}(\sqrt{x-1}+\sqrt{3-x})$

Varsha Aggarwal

Numerade Educator

Problem 79

Find the domain of the function
$f(x)=\log _{1 / 2}\left(\frac{2-x}{x-4}\right)$

Varsha Aggarwal

Numerade Educator

Problem 80

Find the domain of the function
$f(x)=\log _{10}\left(1-\log _{10}\left(x^{2}-5 x+16\right)\right)$

Varsha Aggarwal

Numerade Educator

Problem 81

Find the domain of the function
$f(x)=\log _{10}\left(\left(\log _{10} x^{2}\right)-5 \log _{10} x+6\right)$

Varsha Aggarwal

Numerade Educator

02:12

Problem 82

Find the domain of the function
$f(x)=\log \left(\frac{1}{x-[x]}\right)$

Sahil Kumar

Numerade Educator

01:43

Problem 83

Find the number of real solutions of
$2-x-\log _{e} x=0$

Sahil Kumar

Numerade Educator

Problem 84

Let $f:\{1,2\} \rightarrow\{1,4\}$ and $g:\{1,2\} \rightarrow\{1,4\}$ such
that $f(x)=x^{2} \& g(x)=3 x-2$ Is $f=g$ ?

Varsha Aggarwal

Numerade Educator

00:52

Problem 85

Let $f(x)=x$ and $g(x)=\frac{x^{2}}{x}$ Is $f=g ?$

Varsha Aggarwal

Numerade Educator

Problem 86

Let $f(x)=x \& g(x)=\sqrt{x^{2}}$ Is $f=g ?$

Varsha Aggarwal

Numerade Educator

00:57

Problem 87

Let $f(x)=2 \log x \& g(x)=\log \left(x^{2}\right)$ Is $f=g ?$

Varsha Aggarwal

Numerade Educator

Problem 88

Let $f(x)=\log \left(\frac{x-1}{x-2}\right)$ and
$g(x)=\log (x-1)-\log (x-2) .$ Is $f=g ?$

Varsha Aggarwal

Numerade Educator

Problem 89

Let $f(x)=\sin x$ \& $g(x)=\sqrt{\frac{1-\cos 2 x}{2}}$ Is $f=g ?$

Varsha Aggarwal

Numerade Educator

Problem 90

Let $A=\{1,2,3\}$ and $B=\{4,5\}$
Then find the total number of one-one function between two sets $A$ and $B$.

Varsha Aggarwal

Numerade Educator

Problem 91

Let $A=\{1,2,3\}$ and $B=\{p, q, r\}$
Then find the total number of function between $A$ and $B$.

Varsha Aggarwal

Numerade Educator

01:05

Problem 92

Let $A=\{1,2,3,4\}$ and $B\{a, b, c, d, e\}$
Then find the number of one-one function between $A$ and $B$.

Varsha Aggarwal

Numerade Educator

02:46

Problem 93

Let a function $f: A \rightarrow B$, where $A=\{1,2,3,4\}$ and $B=\{3,4,5,6\}$ such that $f(1)=3$. Then find the number of one-one function between $A$ to $B$.

Sahil Kumar

Numerade Educator

01:49

Problem 94

Let a function $f: A \rightarrow B$, where $A=\{1,2,3\}$ and $B=\{4,5,6\}$ such that $f(1) \neq 4$. Then find the number of one-one function between $A$ to $B$.

Sahil Kumar

Numerade Educator

Problem 95

Let a function $f: A \rightarrow B$, where $A=\{1,2,3\}$ and $B=\{6,7,8\}$
such that $a<b \Rightarrow f(a)<f(b)$,
where $a \in A \& b \in B$.Then find the number of one-one function between $A$ to $B$.

Varsha Aggarwal

Numerade Educator

01:01

Problem 96

Find the number of many-one function between two sets $A=(3,4,5\}$ and $B=\{2,3,4,5\}$.

Varsha Aggarwal

Numerade Educator

Problem 97

Find the number of many-one function between two sets $A=\{1,2,3,4\}$ and $B=\{3,4,5\}$.

Varsha Aggarwal

Numerade Educator

Problem 98

Find the number of many-one function between two sets $A=\{1,2,3\}$ and $B=\{4,5,6\}$ such that $f(1) \neq 4$

Varsha Aggarwal

Numerade Educator

Problem 99

A function $f: R-\{-1\} \rightarrow R-\{1\}$ is defined as $f(x)=\frac{x}{x+1}$. prove that $f$ is onto function.

Varsha Aggarwal

Numerade Educator

Problem 100

Let a function $f: R \rightarrow A$ is defined as $f(x)=\frac{1}{x^{2}+1}$. If $f$ is onto function, then find the set of values of $A .$

Varsha Aggarwal

Numerade Educator

01:15

Problem 101

Let a function $f: A \rightarrow B$ is defined as $f(x)=\frac{x^{2}}{x^{2}+1}$. If $f$ is onto function, then find the set of values of $A \cap B$

Varsha Aggarwal

Numerade Educator

02:11

Problem 102

Find the number of onto function between two sets $A=\{1,2,3,4\}$ and $B=\{4,5\}$

Sahil Kumar

Numerade Educator

02:12

Problem 103

Find the number of onto function between two sets $A=\{1,2,3,4\}$ and $B=\{4,5,6\}$

Sahil Kumar

Numerade Educator

02:46

Problem 104

Find the number of onto function between two sets $A$ and $B$ where $A=\{1,2,3,4,5\}$ and $B=\{6,7,8\}$.

Sahil Kumar

Numerade Educator

01:06

Problem 105

Find the number of into function between two sets $A$ and $B$, where $A=\{1,2,3\}$ and $B=\{4,5\}$

Varsha Aggarwal

Numerade Educator

01:07

Problem 106

Find the number of into function between two sets $A$ and $B$ where $A=\{1,2,3,4\}$ and $B=\{5,6,7\}$

Varsha Aggarwal

Numerade Educator

01:09

Problem 107

Find the number of into functions between two sets $A$ and $B$ where $A=\{1,2,3,4,5\}$ and $B=\{6,7,8\}$.

Varsha Aggarwal

Numerade Educator

01:06

Problem 108

A function $f: R \rightarrow R$ is defined as $f(x)=3 x+5$. Prove that $f$ is a bijective function.

Varsha Aggarwal

Numerade Educator

01:23

Problem 109

A function $f: R^{+} \rightarrow(1, \infty)$ is defined as $f(x)=x^{2}+1$. Prove that the function is bijective.

Varsha Aggarwal

Numerade Educator

01:28

Problem 110

A function $f: R^{+} \rightarrow(0,1)$ is defined as $f(x)=\frac{1}{x^{2}+1}$. Prove that $f$ is a bijective function.

Varsha Aggarwal

Numerade Educator

Problem 111

A function $f: R \rightarrow\left(-\frac{1}{2}, \frac{1}{2}\right)$ is defined as
$f(x)=\frac{x}{x^{2}+1}, \forall x \in(-1,1) .$ Prove that $f(x)$ is a
bijective function.

Varsha Aggarwal

Numerade Educator

01:03

Problem 112

If $A=\{1,2,3,4\}$ and $B=\{2,3,4\}$ then find the number of one-one onto functions between $A \quad f(x)=s i$ and $B$.

Varsha Aggarwal

Numerade Educator

01:00

Problem 113

Find the number of bijective functions between two sets $A$ and $B$, where $A=\{a, b, c, d, e\}$ and $B=\{p$, $q, r, s, t\}$

Varsha Aggarwal

Numerade Educator

Problem 114

Find the number of bijective functions between two sets $A=\{1,2,3,4\}$ and $B=\{5,6,7,8\}$ such that $f(1)=5$

William Mead

Numerade Educator

Problem 115

A function $f: R \rightarrow R$ is defined as $f(x)=3 x+5$. Find $f^{-1}(x) .$

Varsha Aggarwal

Numerade Educator

01:33

Problem 116

A function $f:(0, \infty) \rightarrow(2, \infty)$ is defined as $f(x)=x^{2}$ $+2$. Find $f^{-1}(x)$.

Varsha Aggarwal

Numerade Educator

02:20

Problem 117

A function $f: R^{+} \rightarrow[0,1]$ is defined as $f(x)=\frac{x^{2}}{x^{2}+1}$. Find $f^{-1}(x)$.

Varsha Aggarwal

Numerade Educator

01:28

Problem 118

A function $f:[1, \infty) \rightarrow[1, \infty)$ is defined as
$f(x)=2^{r(x-1)}$. Find $f^{-1}(x)$

Varsha Aggarwal

Numerade Educator

Problem 119

If a function $f$ is bijective such that $f(x)=\frac{10^{x}-10^{-x}}{10^{x}+10^{-x}}$. Find $f^{-1}(x)$

Varsha Aggarwal

Numerade Educator

01:34

Problem 120

A function $f: R \rightarrow R$ is defined as $f(x)=x+\sin x$. Find $f^{-1}(x)$

Rukhmani Jain

Numerade Educator

03:37

Problem 121

A function $f:[2, \infty) \rightarrow[5, \infty)$ is defined as $f(x)=x^{2}-4 x+9 .$ Find its inverse.

Sahil Kumar

Numerade Educator

02:35

Problem 122

Find all the real solutions to the equation
$x^{2}-\frac{1}{4}=\sqrt{x+\frac{1}{4}}$

Varsha Aggarwal

Numerade Educator

01:59

Problem 123

For what values of $m$ is $f(x)=(m+2) x^{3}-3 m x^{2}+9 m$ $x-1$ is invertible?

Varsha Aggarwal

Numerade Educator

01:51

Problem 124

\begin{aligned}
&\text { Let } f(x)=\sqrt[3]{a-x^{3}+3 b x^{2}-3 b x+b^{3}+b} \text {. Find } b \text {, }\\
&\text { if } f(x) \text { is inverse of itself. }
\end{aligned}

Shima Shaw

Numerade Educator

Problem 125

Find the domain of $f(x)=\sin ^{-1}(3 x+4)$

Varsha Aggarwal

Numerade Educator

00:50

Problem 126

Find the domain of $f(x)=\cos ^{-1}(4 x+5)$.

Varsha Aggarwal

Numerade Educator

01:37

Problem 127

Find the domains of
$f(x)=\cos ^{-1}\left(\frac{x}{2}-1\right)+e^{x}+\frac{1}{\lfloor x-1 \mid}$

Varsha Aggarwal

Numerade Educator

Problem 128

Find the domains of
$f(x)=\cos ^{-1}\left(\frac{|x|-3}{5}\right)+\frac{1}{e^{x}+1}$

Varsha Aggarwal

Numerade Educator

Problem 129

Find the domains of
$f(x)=\sin ^{-1}\left(\frac{1-|x|}{2}\right)+\left(\frac{e^{x}-1}{e^{x}+1}\right)+2015$

Varsha Aggarwal

Numerade Educator

01:36

Problem 130

Find the domain of $f(x)=\cos ^{-1}\left(\frac{3}{4+2 \sin x}\right)$

Varsha Aggarwal

Numerade Educator

01:42

Problem 131

Find the domain of
$f(x)=\sin ^{-1}\left(\frac{1+x^{2}}{2 x}\right)+\cos (\sin x)$

Varsha Aggarwal

Numerade Educator

00:54

Problem 132

Find the domains of $f(x)=\sin ^{-1}\left(\log _{2} x\right)$.

Varsha Aggarwal

Numerade Educator

01:30

Problem 133

VFind the domains of $f(x)=\sin ^{-1}\left(\log _{4} x^{2}\right)$.

Varsha Aggarwal

Numerade Educator

Problem 134

Find the domains of
$f(x)=\sin ^{-1}\left(\frac{1}{\mid x^{2}-11}\right)$

Varsha Aggarwal

Numerade Educator

00:51

Problem 135

Let two real functions $f$ and $g$ are defined as $f: R \rightarrow R, f(x)=x^{2}+1$ and $g: R \rightarrow R, f(x)=x-1$
Determine $f_{o} g$ \& $g_{a} f$.

Varsha Aggarwal

Numerade Educator

Problem 136

Find $f_{o} g$ and $g_{o} f$ for the functions $f(x)=\sin x$ and $g(x)=\sqrt{x-2}$

Varsha Aggarwal

Numerade Educator

Problem 137

Find the domain of the function $f_{\infty} g$, where $f(x)=\sqrt{x-3}$ and $g(x)=x^{2}+1$.

Varsha Aggarwal

Numerade Educator

02:31

Problem 138

Find $f_{o} g$, where $f(x)=\sqrt{x}$ and $g(x)=x^{2}-1$.

Sahil Kumar

Numerade Educator

01:40

Problem 139

A function $f(x)$ is defined as $f(x)=\left(a-x^{n}\right)^{1 / n}, x>0, n \in I^{+}$
Then find $f(f(x))+f\left(f\left(\frac{1}{x}\right)\right)$.

Varsha Aggarwal

Numerade Educator

03:36

Problem 140

Find the number of distinct real solutions of the equation $f(f(f(x)))=0$, where $f(x)=x^{2}-1$

Sahil Kumar

Numerade Educator

03:02

Problem 141

$f(x)=\left\{\begin{array}{ll}x^{2} & : x \geq 0 \\ x & : x<0\end{array}\right.$ and $g(x)=-|x|, x \in R$

Sahil Kumar

Numerade Educator

01:47

Problem 142

A function $f$ is defined as
$f(x)=\left\{\begin{array}{ll}1+x: & x \geq 0 \\ x-x: & x<0\end{array}\right.$. Then find $f_{a} f$

Sahil Kumar

Numerade Educator

03:38

Problem 143

Let $f: R^{+} \rightarrow R$ be defined as $f(x)=x^{2}-x+2$ and $g:[1,2] \rightarrow[1,2]$ be defined as $g(x)=\{x\}+1$, where $\{,\}=$, Fractional part function. If the domain and range of $f(g(x))$ are $[a, b]$ and $[c, d)$, then find the value of $\frac{b}{a}+\frac{d}{c}$

Sahil Kumar

Numerade Educator

Problem 144

If $g$ is the inverse of $f$ and $f^{\prime}(x)=\sin x$, find $g^{\prime}(x)$.

Varsha Aggarwal

Numerade Educator

01:48

Problem 145

If $g(x)=\frac{1}{1-x}$ and $g_{2}(x)=g(g(x))$ and $g_{3}(x)=g(g(g(x)))$,
then find the range of $g_{2016}(x)$.

Varsha Aggarwal

Numerade Educator

03:24

Problem 146

If the roots of $(c-1)\left(x^{2}+x+1\right)^{2}-(c+1)\left(x^{4}+x^{2}+1\right)=0$ are
real and distinct and $f(x)=\frac{1-x}{1+x}$, then find the value of $f(f(x))+f\left(f\left(\frac{1}{2}\right)\right)$.

Shima Shaw

Numerade Educator

03:58

Problem 147

If $f(x)=\sin ^{2} x+\sin ^{2}\left(x+\frac{\pi}{3}\right)+\cos x \cos \left(x+\frac{\pi}{3}\right)$
and $g\left(\frac{5}{4}\right)=1$, then find $\left(g_{0} f\right)(x)$

Sahil Kumar

Numerade Educator

Problem 148

Let $f(x)=1+x^{2}$. Find a function $g(x)$ such that $f(g(x))=1+x^{2}-2 x^{3}+x^{4}$

Varsha Aggarwal

Numerade Educator

Problem 149

Let $g(x)=1+x-[x]$ and $f(x)=\left\{\begin{aligned}-1 &: x<0 \\ 0 &: x=0 \\ 1 &: x>0 \end{aligned}\right.$ then for all $x$, find $f(g(x))$

Varsha Aggarwal

Numerade Educator

01:21

Problem 150

Let $f(x)=\frac{\alpha x}{x+1}, x \neq-1$ then for what value of $\alpha$ is
$f(f(x))=x ?$

Varsha Aggarwal

Numerade Educator

Problem 151

$f(x)=x\left(\frac{3^{x}-1}{3^{x}+1}\right)$

Varsha Aggarwal

Numerade Educator

00:58

Problem 152

$f(x)=x \sin \left(x^{2}+1\right)$

Varsha Aggarwal

Numerade Educator

Problem 153

$f(x)=\tan ^{-1}\left(\sin \left(\cos ^{-1}(x)\right)\right)$

Varsha Aggarwal

Numerade Educator

Problem 154

$f(x)=\sin x+\cos x$

Varsha Aggarwal

Numerade Educator

01:36

Problem 155

$f(x)=\left(\frac{x}{e^{x}-1}+\frac{x}{2}+1\right)$

Varsha Aggarwal

Numerade Educator

Problem 156

$f(x)=\log \left(\frac{3-x}{3+x}\right)$

Varsha Aggarwal

Numerade Educator

01:36

Problem 157

$f(x)=\log \left(x+\sqrt{x^{2}+1}\right)$

Varsha Aggarwal

Numerade Educator

01:17

Problem 158

$f(x)=\sqrt{x^{2}-x+1}-\sqrt{x^{2}+x+1}$

Varsha Aggarwal

Numerade Educator

01:51

Problem 159

Determine the nature of the function
$f(x)=x \sin ^{2} x+\tan \left(x^{5}\right)+\log \left(\frac{2-x}{2+x}\right)$

Varsha Aggarwal

Numerade Educator

01:06

Problem 160

Determine the nature of the function
$f(x)=\left(\tan \left(x^{5}\right)\right) e^{x^{2} \operatorname{sen}\left(x^{7}\right)}$

Varsha Aggarwal

Numerade Educator

Problem 161

Determine the nature of the function
$f(x)=\frac{x^{2}\left(\log \left(x+\sqrt{x^{2}+1}\right)\right)}{\left(x \cos ^{2} x+\tan x+x^{7}\right)}$

Varsha Aggarwal

Numerade Educator

Problem 162

Determine the nature of the function
$f(x)=\frac{\sin \left(\tan \left(\log \left(x+\sqrt{x^{2}+1}\right)\right)\right)}{\sqrt{x^{2}+1}+\sin (\cos x)+\cos (\sin x)}$

Varsha Aggarwal

Numerade Educator

01:20

Problem 163

Express the function $f(x)=4^{\sin x}$ as a sum of an even and an odd function.

Varsha Aggarwal

Numerade Educator

01:25

Problem 164

Express the function $f(x)=(1+x)^{2015}$ as a sum of even and an odd functi-

Varsha Aggarwal

Numerade Educator

02:08

Problem 165

Let $f(x)= \begin{cases}x+x^{2} & : 0 \leq x<3 \\ x+x & : & 3<x \leq 5\end{cases}$
Then find its even and odd extension.

Varsha Aggarwal

Numerade Educator

01:25

Problem 166

Let $f(x)=x+e^{x}+\sin x$ be defined on $[0,2]$. Then find its even and odd extension.

Varsha Aggarwal

Numerade Educator

01:23

Problem 167

Let $f(x)=x^{2}+x+\sin x-\cos x+\log _{e}(1+x)$ be
defined on $[0,1]$. Find its even and odd extension in the interval $[-1,1]$

Varsha Aggarwal

Numerade Educator

Problem 168

Find the period of $f(x)=3 \sin 4 x+4 \cos 3 x$

Varsha Aggarwal

Numerade Educator

01:16

Problem 169

Find the period of $f(x)=3 \sin 4 x+4 \operatorname{lsin} 4 x \mid$

Varsha Aggarwal

Numerade Educator

Problem 170

Find the period of $f(x)=\sin x \cdot \operatorname{cosec} x$

Varsha Aggarwal

Numerade Educator

Problem 171

Find the period of $f(x)=\tan x \cdot \cot x$.

Varsha Aggarwal

Numerade Educator

01:31

Problem 172

Find the period of $f(x)=5 \sin (2 \sqrt{2} x)+7 \cos (3 \sqrt{2} x)$

Varsha Aggarwal

Numerade Educator

01:58

Problem 173

Find the period of
$f(x)=3 \sin (2 \sqrt{3} x)+2 \cos (5 \sqrt{3} x)$

Varsha Aggarwal

Numerade Educator

Problem 174

Find the period of
$f(x)=\sin ^{2} x+\sin ^{2}\left(x+\frac{\pi}{3}\right)-\cos x \cos \left(x+\frac{\pi}{3}\right)$

Varsha Aggarwal

Numerade Educator

01:16

Problem 175

Find the period of $f(x)=\frac{|\sin x+\cos x|}{\sin x|+| \cos x \mid}$

Varsha Aggarwal

Numerade Educator

00:57

Problem 176

Find the period of $f(x)=\frac{|\sin x+\cos x|}{|\sin x|-|\cos x|}$

Varsha Aggarwal

Numerade Educator

01:00

Problem 177

Find the period of $f(x)=\frac{1}{2}\left(\frac{|\sin x|}{\cos x}+\frac{\sin x}{|\cos x|}\right)$

Varsha Aggarwal

Numerade Educator

Problem 178

Find the period of $f(x)=3 \sin \{2 x\}+2 \cos \{3 x\}$

Varsha Aggarwal

Numerade Educator

Problem 179

Let $f(x)+f(x+3)=5$ for all $x$ in $R$. Prove that $f(x)$ is periodic with period 6 .

Varsha Aggarwal

Numerade Educator

02:37

Problem 180

Find the period of the function $f^{\prime}$ which satisfy the equation $f(x+4)+f(x-4)=f(x)$

Varsha Aggarwal

Numerade Educator

03:11

Problem 181

If $f$ is a function satisfying the equation $f(x-1)+f(x+1)=\sqrt{2} f(x)$, then prove that $f(x)$ is periodic with period 8 .

Varsha Aggarwal

Numerade Educator

01:44

Problem 182

Let $f(x+1)+f(x+5)=f(x+3)+f(x+7)$ for
all $x$ in $R$. Prove that $f(x)$ is periodic Functional Equation.

Varsha Aggarwal

Numerade Educator

01:27

Problem 183

Let $f: R-\{0\} \rightarrow R$ be any real function such that $f(x)+3 f\left(\frac{1}{x}\right)=5 x .$ Find $f(x)$.

Varsha Aggarwal

Numerade Educator

02:03

Problem 184

Let $2 f(\sin x)+3 f(\cos x)=5, \forall x \in R .$ Find $f(x)$

Varsha Aggarwal

Numerade Educator

02:17

Problem 185

CLet a function $f$ saisfy $f(x+1)=f(x)+x, \forall x \in N$ where $f(1)=0$. Find $f(x)$.

Sahil Kumar

Numerade Educator

00:57

Problem 186

Suppose $f$ is real function satisfying $f(x+f(x))=4 f(x)$ and $f(1)=4$ Then find $f(5)$.

Sahil Kumar

Numerade Educator

02:35

Problem 187

Find all polynomial $P(x)$ which satisfy the relation $P(x+1)=P(x)+2 x+1$, where $P(0)=0$

Sahil Kumar

Numerade Educator

03:28

Problem 188

Find the natural number ' $a$ ' for which
$\sum_{k=1}^{n} f(a+k)=16\left(2^{n}-1\right)$, where $f$ satisfies the
relation $f(x+y)$
$=f(x) \cdot f(y)$ for all $x, y$ are in $N$ and $f(1)=2$

Shima Shaw

Numerade Educator

02:37

Problem 189

Let a polynomail function $f$ satisfies the relation $f(x)+f\left(\frac{1}{x}\right)=f(x) \cdot f\left(\frac{1}{x}\right)$ for all $x \in R-\{0\}$
Then find $f(x)$.

Shima Shaw

Numerade Educator

01:19

Problem 190

If $f(x)$ is a polynomial function of $x$, satisfying $f(x) \cdot f(y)=f(x)+f(y)+f(x y)-2$ such that $f(1)=2$
and $f(2)=5$, then find the value of $f(6)+2009$.

Varsha Aggarwal

Numerade Educator