Properties of Composite Function

Q. If $a^3=11\sqrt{2}+9\sqrt{3},$ then $a$ is

1. $\sqrt{3}+\sqrt{2}$
2. $\sqrt{2}+\sqrt{6}$
3. $\sqrt{3}+\sqrt{6}$
4. $\sqrt{3}+\sqrt{5}$

Q.

If $x-\frac{1}{x}=8$ find the value of $x^2+\frac{1}{x^2}$ and $x^4+\frac{1}{x^4}$

Q. In an open interval $(0,\frac{\pi }{2})$,

1. $\cos x+x\sin x<1$
2. $\cos x+x\sin x<\frac{1}{2}$
3. $\cos x+x\sin x>1$
4. no specific order relation can be ascertained between $\cos x+x\sin x$ and $1$

Q.

The function $f$ satisfies the functional equation $3f\left(x\right)+2f\left(\frac{x+59}{x-1}\right)=10x+30$ for all real $x\ne 1$. Then value of $f\left(7\right)$ is:

1. $8$

2. $4$

3. $-8$

4. $11$

5. $44$

Q. Find the principal value of ${\tan }^{-1}(-\sqrt{3}).$

Q.

If $\tan \left(\mathrm{A}\right)=\sqrt{3}$ find values of all other trigonometric ratios.

Q. Let $f\left(x\right)=\frac{x}{e^x-1}+\frac{x}{2}+1$ and $g$ is either odd or even function. If $fog$ is defined, then

1. $fog$ is odd function
2. $fog$ is even function
3. $f\left(x\right)$ is even function
4. $f\left(x\right)$ is odd function

Q. Let $g:R\to R$ be a differentiable function with $g\left(0\right)=0,g^{\prime }\left(0\right)=0\text{and}{g}^{\prime }\left(1\right)\ne 0.$ Let
$f\left(x\right)=\{\begin{array}{c}\frac{x}{|x|}g\left(x\right),x\ne 0\\ 0,x=0\end{array}$
and $h\left(x\right)=e^{|x|}$ for all $x\in R.$ Let $(f\circ h)\left(x\right)$ denotes $f\left(h\right(x\left)\right)$ and $(h\circ f)\left(x\right)$ denotes $h\left(f\right(x\left)\right).$ Then which of the following is (are) true?

1. $f$ is differentiable at $x=0$
2. $h$ is differentiable at $x=0$
3. $f\circ h$ is differentiable at $x=0$
4. $h\circ f$ is differentiable at $x=0$

Q. Let $J=\underset{0}{\overset{1}{\int }}\frac{x}{1+x^8}dx.$
Consider the following assertions:
I. $J>\frac{1}{4}$

II. $J<\frac{\pi }{8}$
Then

1. Only I is true
2. Only II is true
3. Both I and II are true
4. neither I nor II is true

Q. Let $g_i:[\frac{\pi }{8},\frac{3\pi }{8}]\to R,i=1,2$ and $f:[\frac{\pi }{8},\frac{3\pi }{8}]\to R$ be functions such that $g_1\left(x\right)=1,g_2\left(x\right)=|4x-\pi |$ and $f\left(x\right)={\sin }^{2}x,$ for all $x\in [\frac{\pi }{8},\frac{3\pi }{8}].$
If $S_i=\underset{\pi /8}{\overset{3\pi /8}{\int }}f\left(x\right)\cdot g_i\left(x\right)dx,i=1,2,$ then the value of $\frac{48S_2}{{\pi }^2}$ is

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

Q. Let $f,g:R\to R$ be two function defined as $f\left(x\right)=|x|+x$ and $g\left(x\right)=|x|-x$, for all $xϵR$. Then find $fog$, hence find $fog\left(5\right),fog(-3)$.

Q. If $f\left(x\right)=\sqrt{x+3}$ and $g\left(x\right)=x^2+1$, then $f\left(g\right(x)$ =

1. $\sqrt{x^2+4}$
2. $x+4$
3. $x+4$
4. $\sqrt{x^2+3}$

Q. If $y={\tan }^{-1}\left(\frac{\log \left(\frac{e}{x^2}\right)}{\log \left(e{x}^2\right)}\right)+{\tan }^{-1}\left(\frac{3+2\log x}{1-6\log x}\right)$, then $\frac{d^{2}y}{d{x}^2}$ is

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

Q. $tan\frac{2\pi }{5}-tan\frac{\pi }{15}-\sqrt{3}tan\frac{2\pi }{5}tan\frac{\pi }{15}$ is equal to

1. $1$
2. $\frac{1}{\sqrt{3}}$
3. $-\sqrt{3}$
4. $\sqrt{3}$

Q. $\tan \frac{\pi }{11}+\tan \frac{2\pi }{11}+\tan \frac{4\pi }{11}+\tan \frac{7\pi }{11}+\tan \frac{9\pi }{11}+\tan \frac{10\pi }{11}$.

Q. If $f\left(x\right)=4x^2-1$ and $g\left(x\right)=8x+7,g^{o}f\left(2\right)=$

1. 15
2. 23
3. 127
4. 345
5. 2115

Q. $\text{If}f\left(x\right)=\left|\begin{array}{ccc}1+x^n& (1-x{)}^n& 2+x^n\\ (2+x{)}^n& (2+x{)}^n& 1\\ (3-x{)}^n& 1& 3+x\end{array}\right|$,
$\text{then the constant term in the expansion is}$

1. $(3^n-1)(1-2^{n+1})$
2. $3^n(1-2^{n+1}-2^{n+1}-1)$
3. $(3^n-1)(2^{n+1}-1)$
4. $2^{n+1}(1-3^n)+(3^n+1)$

Q. If f is a differentiable function on R and $f^{\prime }\left(0\right)=2$ satisfying $f(x+y)=\frac{f\left(x\right)+f\left(y\right)}{1-f\left(x\right)f\left(y\right)}$ then $f\left(\pi /8\right)$ is equal to

1. $1/2$
2. $1$
3. $3/2$
4. $\tan \pi /8$

Q. If $|x-2|\le 1,$ then

1. $x\in [1,3]$
2. $x\in (1,3)$
3. $x\in [-1,3)$
4. $x\in (-1,3)$

Q.

Find the value of each of the following:

$$\begin{gathered} \left(i\right){\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right) \\ \left(ii\right){\tan }^{-1}\left(\frac{-1}{\sqrt{3}}\right) \\ \left(iii\right){\tan }^{-1}\left(\cos \left(\frac{\mathrm{\pi }}{2}\right)\right) \\ \left(iv\right){\tan }^{-1}\left(2\cos \left(\frac{2\mathrm{\pi }}{3}\right)\right) \end{gathered}$$

Q. Let $f\left(x\right)=|x-2|$ and $g\left(x\right)=f\left(f\right(x\left)\right),x\in [0,4].$ Then ${\int }_0^3\left(g\right(x)-f(x\left)\right)dx$ equal to

1. $\frac{1}{2}$
2. $0$
3. $1$
4. $\frac{3}{2}$

Q. If $f:R^{+}\to R^{+}$ is a polynomial function satisfying the functional equation $f\left(f\right(x\left)\right)=6x-f\left(x\right),$ then $f\left(17\right)$ is equal to

1. $51$
2. $34$
3. $17$
4. $-51$

Q.

If $f\left(x\right)=\sqrt{x+3}$ and $g\left(x\right)=1+x^2,$ then $fog\left(x\right)=$ ____.

1. $x+4$

2. $\sqrt{x^2+4}$

3. $x+3$

4. $\sqrt{x^2-4}$

Q. Let $f\left(x\right)=\sin \left(\frac{\pi }{6}\sin \left(\frac{\pi }{2}\sin x\right)\right)$ for all $x\in R$ and $g\left(x\right)=\frac{\pi }{2}\sin x$ for all $x\in R.$ Let $(f\circ g)\left(x\right)$ denote $f\left(g\right(x\left)\right)$ and $(g\circ f)\left(x\right)$ denotes $g\left(f\right(x\left)\right).$ Then which of the following is (are) true?

1. Range of $f$ is $[\frac{-1}{2},\frac{1}{2}]$
2. Range of $f\circ g$ is $[\frac{-1}{2},\frac{1}{2}]$
3. $\underset{x\to 0}{lim}\frac{f\left(x\right)}{g\left(x\right)}=\frac{\pi }{6}$
4. There is an $x\in R$ such that $(g\circ f)\left(x\right)=1$

Q. Find the composition function $(f\circ g)\left(x\right)$ and its domain.
$f\left(x\right)=\sqrt{(}-x+1),g(x)=x^2-8$

1. $[-2,2]$
2. $[-1,6]$
3. $[-3,3]$
4. $[-\infty ,\infty ]$

Q. If $f\left(x\right)=2,g\left(x\right)=x^2,h\left(x\right)=2x,\forall x\in R$ then find $\left(f\right(g\left(h\right(x\left)\right)\left)\right)$

Q. $f\left(x\right)=2x+1;$ $g\left(x\right)=x^2-1$
$(f\circ g)\left(x\right)=$

1. $x^2+2x$
2. $2x^3+x^2-2x-1$
3. $2x^2-1$
4. $4x^2+4x$
5. $2(x^2+x+1)$

Q. Find the composition function $(f\circ g)\left(x\right)$ and its domain.
$f\left(x\right)=x^2+1,g\left(x\right)=\sqrt{2x}$

1. $[1,+\infty )$
2. $[0,+\infty )$
3. $[-\infty ,+\infty )$
4. $[-1,+\infty )$

Q. The value of determinant $\Delta =\left|\begin{array}{ccc}a& b& c\\ a+2x& b+2y& c+2z\\ x& y& z\end{array}\right|$ is

1. $xyz$
2. $abc$
3. $2xyz+abc$
4. $0$

Q. $\frac{d}{dx}\left({\sec }^{-1}x\right)=$ ?