Composite Function

Q. For $x\in (0,\frac{3}{2}),$ let $f\left(x\right)=\sqrt{x},g\left(x\right)=\tan x$ and $h\left(x\right)=\frac{1-x^2}{1+x^2}.$ If $\varphi \left(x\right)=\left(\right(hof\left)og\right)\left(x\right),$ then $\varphi \left(\frac{\pi }{3}\right)$ is equal to

1. $\tan \left(\frac{\pi }{12}\right)$
2. $\tan \left(\frac{5\pi }{12}\right)$
3. $\tan \left(\frac{7\pi }{12}\right)$
4. $\tan \left(\frac{11\pi }{12}\right)$

Q. Let $f:R\to R$ be a differentiable function with $f\left(0\right)=1$ and satisfying the equation
$f(x+y)=f\left(x\right)f^{\prime }\left(y\right)+f^{\prime }\left(x\right)f\left(y\right)$ for all $x,y\in R.$
Then, the value of ${\log }_e\left(f\right(4\left)\right)$ is

Q. Let there are $n$ number of polynomial functions $f_i:R\to R,i\in N$ satisfying the equation
$f(a+f(a+b\left)\right)+f\left(ab\right)-f(a+b)-bf\left(a\right)=a$ $\forall a,b\in R$. Then

1. there exists at least one odd function.
2. there exists at least one even function.
3. $f_i\left(1\right)=1\forall i\in N$
4. $\sum _{i=1}^n{f}_i\left(x\right)=k$, where $k$ is a constant.

Q.

The number of solutions of the equation $|x-1|=e^x$ is

1. $0$

2. $1$

3. $2$

4. $3$

Q. Let$f\left(x\right)={\log }_e\left(\sin x\right),(0<x<\pi )$ and $g\left(x\right)={\sin }^{-1}\left(e^x\right),(x\ge 0).$If $\alpha$ is a positive real number such that $a=\left(fog{)}^{\prime }\right(\alpha )$ and $b=\left(fog\right)\left(\alpha \right),$ then :

1. $a{\alpha }^2-b\alpha -a=0$
2. $a{\alpha }^2+b\alpha +a=0$
3. $a{\alpha }^2+b\alpha -a=-2{\alpha }^2$
4. $a{\alpha }^2-b\alpha -a=1$

Q. Let $f\left(x\right)=x^2,x\in R.$ For any $A\subseteq R$, define $g\left(A\right)=\{x\in R:f(x)\in A\}$. If $S=[0,4]$, then which one of the following statements is not true ?

1. $f\left(g\right(S\left)\right)\ne f\left(S\right)$
2. $f\left(g\right(S\left)\right)=S$
3. $g\left(f\right(S\left)\right)\ne S$
4. $g\left(f\right(S\left)\right)=g\left(S\right)$

Q. Let$f\left(x\right)={\log }_e\left(\sin x\right),(0<x<\pi )$ and $g\left(x\right)={\sin }^{-1}\left(e^x\right),(x\ge 0).$If $\alpha$ is a positive real number such that $a=\left(fog{)}^{\prime }\right(\alpha )$ and $b=\left(fog\right)\left(\alpha \right),$ then :

1. $a{\alpha }^2-b\alpha -a=0$
2. $a{\alpha }^2+b\alpha +a=0$
3. $a{\alpha }^2+b\alpha -a=-2{\alpha }^2$
4. $a{\alpha }^2-b\alpha -a=1$

Q. Let $f\left(x\right)=\frac{3x}{4}+1,f_2\left(x\right)=f\left(f\right(x\left)\right)$ and for $n\ge 2,f_{n+1}\left(x\right)=f\left(f_n\right(x\left)\right)$. If $\alpha =\underset{n\to \infty }{lim}{f}_n\left(x\right)$. Then

1. $a$ has $9$ integral solutions for $|a-2|\le \alpha$
2. $\underset{0}{\overset{2/3}{\int }}{f}_n\left(x\right)dx<0$
3. the line $8y=\alpha$ intercepts a chord of length of $\sqrt{3}$ with $x^2+y^2=1$
4. $\alpha$ is dependent on $x$

Q. If $f\left(x\right)=\{\begin{array}{c}x,x\text{is rational}\\ 1-x,x\text{is irrational}\end{array}$,
then $f\left(f\right(x\left)\right)$ is

1. $x\forall x\in R$
2. $f\left(x\right)=\{\begin{array}{c}x,x\text{is rational}\\ 1-x,x\text{is irrational}\end{array}$
3. $f\left(x\right)=\{\begin{array}{c}x,x\text{is irrational}\\ 1-x,x\text{is rational}\end{array}$
4. None of these

Q. If $f\left(x\right)={\log }_e\left(\frac{1-x}{1+x}\right),|x|<1$, then $f\left(\frac{2x}{1+x^2}\right)$ is equal to:

1. $2f\left(x^2\right)$
2. $-2f\left(x\right)$
3. $2f\left(x\right)$
4. $\left(f\right(x){)}^2$

Q. Let $f:R\to R$ be a differentiable function with $f\left(0\right)=1$ and satisfying the equation $f(x+y)=f\left(x\right)f^{\prime }\left(y\right)+f^{\prime }\left(x\right)f\left(y\right)$ for all $x,y\in R.$ Then the value of ${\log }_e\left(f\right(4\left)\right)$ is

Q.

Let $f\left(x\right)=x^2$ and $g\left(x\right)=\sin x$ for all $x\in R$. Then the set of all $x$ satifying $(f\circ g\circ g\circ f)\left(x\right)=(g\circ g\circ f)\left(x\right)$, where $(f\circ g)\left(x\right)=f\left(g\right(x\left)\right)$, is

1. $\pm \sqrt{n\pi ,}n\in 0,1,2,...$

2. $\pm \sqrt{n\pi ,}n\in 1,2,...$

3. $\frac{\pi }{2}+2n\pi ,n\in ...-2,1,0,1,\mathrm{2...}$

4. $2n\pi ,n\in ...-2,-1,0,1,2,....$

Q. The point of inflection for $y=f\left(x\right)=x{e}^x$ is:

1. $0$
2. $-2$
3. $2$
4. $-1$

Q. Let f and g be real functions defined by $f\left(x\right)=\sqrt{x+2}\mathrm{and}g\left(x\right)=\sqrt{4-x^2}.\mathrm{Then},\mathrm{the}\mathrm{domain}\mathrm{of}\mathrm{the}\mathrm{function}\left(\frac{f}{g}\right)\left(x\right)\mathrm{is}$

1. $\left(-2,2\right]$
2. $\left(-2,2\right)$
3. $[-2,2)$
4. $(-2,2]$

Q. Let $f:\{2,3,4,5\}\to \{3,4,5,9\}$ and $g:\{3,4,5,9\}\to \{7,11,15\}$ be functions defined as $f\left(2\right)=3,f\left(3\right)=4,f\left(4\right)=f\left(5\right)=5$ and $g\left(3\right)=g\left(4\right)=7$ and $g\left(5\right)=g\left(9\right)=11.$ Then $gof\left(5\right)$ is

Q. Let $f\left(x\right)=x^2+20x+90$, then the number of real solutions of the equation f(f(f(x))) = 0 is

1. 6
2. 4
3. 2
4. \N

Q.

Let $F\left(x\right)=\frac{x}{(1+x^n{)}^{1/n}}$ for $n\ge 2$ and $g\left(x\right)=\underset{\text{f occours n times}}{\underset{}{(f\circ f\circ ..\circ f)}}\left(x\right)$. Then $\int x^{x-2}g\left(x\right)dx$ equals.

1. $\frac{1}{n(n-1)}(1+n{x}^n{)}^{1-\frac{1}{n}}+K$
   

2. $\frac{1}{n-1}(1+n{x}^n{)}^{1-\frac{1}{n}}+k$
   

3. $\frac{1}{n(n+1)}(1+n{x}^n)1+\frac{1}{n}+K$
   

4. $\frac{1}{n+1}(1+n{x}^n{)}^{1+\frac{1}{n}}+K$

Q. Let there are $n$ number of polynomial functions $f_i:R\to R,i\in N$ satisfying the equation
$f(a+f(a+b\left)\right)+f\left(ab\right)-f(a+b)-bf\left(a\right)=a$ $\forall a,b\in R$. Then

1. there exists at least one odd function.
2. there exists at least one even function.
3. $f_i\left(1\right)=1\forall i\in N$
4. $\sum _{i=1}^n{f}_i\left(x\right)=k$, where $k$ is a constant.

Q. If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$, then ${\cos }^{-1}x+{\cos }^{-1}y$ is equal to

1. $\frac{\pi }{4}$
2. $\pi$
3. $\frac{\pi }{2}$
4. $\frac{3\pi }{4}$

Q. For $x\in R-\{0,1\},$ let $f_1\left(x\right)=\frac{1}{x}$, $f_2\left(x\right)=1-x$ and $f_3\left(x\right)=\frac{1}{1-x}$ be three given functions. If a function, $J\left(x\right)$ satisfies $\left(f_{2}oJo{f}_1\right)\left(x\right)=f_3\left(x\right)$, then $J\left(x\right)$ is equal to :

1. $f_1\left(x\right)$
2. $f_2\left(x\right)$
3. $f_3\left(x\right)$
4. $\frac{1}{x}\cdot f_3\left(x\right)$

Q. $f:(0,\infty )\to (0,\infty ),f\left(xf\right(y\left)\right)=x^2{y}^a(a\in R)$ then

Number of solutions of $2f\left(x\right)=e^x$ is

1. \N
2. 1
3. 2
4. 3

Q. Let $f\left(x\right)=\frac{8}{x},x\ne 0\text{and}fog\left(x\right)=4x.$ If $gof\left(x\right)=3,$ then $x$ is

Q. If $f:R\to R$ and $g:R\to R$ are given by f(x) = |x| and g(x) = [x], then $g\left(f\right(x\left)\right)\le f\left(g\right(x)$ is true for -

1. $Z\cup (-\infty ,0)$
2. $(-\infty ,0)$
3. $Z$
4. $R$

Q. Consider the graph of a real-valued continuous function $f\left(x\right)$ defined on $R$ (the set of real numbers) as shown below.


The number of real solutions of the equation $f\left(f\right(x\left)\right)=4$ is