Inverse of a Function

Q.

If $g\left(x\right)=x^2+x-1$ and $\left(gof\right)\left(x\right)=4x^2+10x+5$, then $f\left(\frac{5}{4}\right)$ is equal to

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

2. $\frac{-1}{2}$

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

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

Q. If $f\left(x\right)=sinx+cosx,g\left(x\right)=x^2-1$ then $g\left(f\right(x\left)\right)$ in invertible in the Domain

1. 
2. 
3. 
4. 

Q.

Let f$f:R\to R$and $g:R\to R$ be function sastisfying $f(x\hspace{0.17em}+y)=f\left(x\right)+f\left(y\right)+f\left(x\right).f\left(y\right)$and $f\left(x\right)=xg\left(x\right)$ for all $x,y\in R.$ If $\underset{x\to 0}{\lim }g\left(x\right)=1$, then which of the following statements is/are True?

1. $f$ is differentiable at every $x\in R$

2. If $g\left(0\right)=1$, then g is differentiable at every $x\in R$

3. The derivative $\left(1\right)$is equal to $1$

4. The derivative $\left(0\right)$is equal to $1$

Q. The number of real solutions of the equation $e^{4x}+4e^{3x}-58e^{2x}+4e^x+1=0$ is

Q.

The relation R is defined on the set of natural numbers as $\left\{(a,b):a=2b\right\}$. Then $R^{-1}$ is given by

1. $\left\{(2,1),(4,2),(6,3),\dots \right\}$

2. $\left\{(1,2),(2,4),(3,6),\dots \right\}$

3. $R^{-1}$ is not defined

4. None of the above

Q. If domain of the function $f\left(x\right)=x^2-6x+7$ is $(-\infty ,\infty )$ then its range is

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

Q.

If $x^2+4y^2-8x+12=0$ is satisfied by real values of x and y then 'y' $ϵ$

1. [-2, -1]

2. [2, 6]

3. [2, 5]

4. [-1, 1]

Q.

Consider $f:R\to R$ given by f(x)=4x+3. Show that f is invertible. Find the inverse of f.

Q. If f (x) +f (x+a) + f(x+2a) +....+ f (x+na) = constant; $\forall x$ $ϵR$ and $a>0$ and f(x) is periodic, then period of f(x), is

1. na
2. (n+1) a
3. 
4. e^{na}

Q. Let $f:[a,\infty )\to [a,\infty )$ be defined by $f\left(x\right)=x^2-2ax+a(a+1)$. If one of the solutions of the equation $f\left(x\right)=f^{-1}\left(x\right)$ is $5049$, then the other solution can be

1. $5051$
2. $5048$
3. $5052$
4. $5050$

Q.

If the function $f\left(x\right)=\frac{\left[\cos \left(\sin x\right)-\cos x\right]}{x^4}$ is continuous at each point in its domain and $f\left(0\right)=\frac{1}{k}$, then $k$ is:

Q.

A relation $R$ is defined over the set of non-negative integers as $xRy\Rightarrow x^2+y^2=36$. What is $R$?

1. $\left\{(0,6)\right\}$

2. $\left\{(6,0)(\surd 11,5),(3,3\surd 3)\right\}$

3. $\left\{(6,0),(0,6)\right\}$

4. $\left\{(\surd 11,5),(2,4\surd 2),(5,\surd 11),(4\surd 2,2)\right\}$

Q. Inverse exists for a function which is

1. Injective
2. Surjective
3. Bijective
4. Many-one

Q. If the value of x is so small that $x^2$ and greater powers can be neglected, then $\frac{\sqrt{1+x}+\sqrt[3]{3(1-x{)}^2}}{1+x+\sqrt{1+x}}$ is equal to

1. $1+\frac{5}{6}x$
2. $1-\frac{5}{6}x$
3. $1+\frac{2}{3}x$
4. $1-\frac{2}{3}x$

Q.

$\frac{1-{\tan }^2(45°-A)}{1+{\tan }^2(45°-A)}$ is equal to

1. $\sin 2A$

2. $\cos 2A$

3. $\tan 2A$

4. $cot2A$

Q.

If $f\left(x\right)=\frac{(4x+3)}{(6x-4)},x\ne \frac{2}{3}$ show that (fof)(x)=x, for all $x\ne \frac{2}{3}$.What is the inverse of f?

Q.

The function $y=a(1–\cos x)$ is maximum when $x=$

1. $\mathrm{\pi }$

2. $\frac{\mathrm{\pi }}{2}$

3. $-\frac{\mathrm{\pi }}{2}$

4. $-\frac{\mathrm{\pi }}{6}$

Q.

If $f:Q\to Q$ is defined as $f\left(x\right)=x^2$, then $f^{-1}\left(9\right)$ is equal to

1. - 3

2. 3

3. $\varphi$

4. {-3, 3}

Q. If $f:R\to R$ is an invertible function such that $f\left(x\right)$ and $f^{-1}x$ are symmetric about the line $y=-x$, then

1. $f\left(x\right)$ is odd
2. $f\left(x\right)$ and $f^{-1}\left(x\right)$ may not be symmetric about the line $y=x$
3. $f\left(x\right)$ may not be odd
4. None of these

Q.

How to find the inverse of a matrix?

Q.

Check the injectivity and surjectivity of the following functions:
$f:Z\to Z$ given by $f\left(x\right)=x^2$

Q.

The value of $\left(\text{Z+3)}\right(\overline{\text{Z}}+3)$ is equivalent to :

Q.

If $f:R\to R$ be defined by $f\left(x\right)=x^2+1$, then find $f^{-1}\left\{17\right\}$ and $f^{-1}\{-3\}$.

Q.

Which of the following functions are periodic?

1. $f\left(x\right)=\log \left(x\right),x>0$

2. $f\left(x\right)=e^x,x\in R$

3. $f\left(x\right)=x–\left[x\right],x\in R$

4. $f\left(x\right)=x+\left[x\right],x\in R$

Q. If the equation $a{x}^n+3x^5+b{x}^2+c=0,$ $n\in Z^{+}$ has infinite number of real solutions, then $a+b+c+n$ is equal to

1. $2$
2. $3$
3. $4$
4. $6$

Q. If $A=\left[\begin{array}{cc}ab& b^2\\ -a^2& -ab\end{array}\right]$ and $A^n=O,$ then the minimum value of ${}^{\prime }{n}^{\prime }$ is

Q. If ${}^{\prime }{X}^{\prime }$ has a binomoial distribution with parameters $n=6,p$ and $P(X=2)=12,P(X=3)=5$ then $p=$

1. $\frac{1}{2}$
2. $\frac{5}{12}$
3. $\frac{5}{16}$
4. $\frac{5}{21}$

Q. The maximum possible domain $D_f$ and the corresponding range $R_f$ of $f\left(x\right)=(-1{)}^x$ are

1. $D_f=R$
2. $R_f=\{-1,1\}$
3. $D_f=Z$
4. $R_f=[-1,1]$

Q. The domain of ${{}^{9-x}P}_{x-2}$ is

1. $x\in \{3,4,5\}$
2. $x\in \{2,3,4,5\}$
3. $x\in \{2,3,4,5,6\}$
4. $x\in \{3,4,5,6\}$

Q. Let $f:[a,\infty )\to [a,\infty )$ be defined by $f\left(x\right)=x^2-2ax+a(a+1)$. If one of the solutions of the equation $f\left(x\right)=f^{-1}\left(x\right)$ is $5049$, then the other solution can be

1. $5051$
2. $5048$
3. $5052$
4. $5050$