Inverse of a Function

Q.

If $R$ is a relation from $\{11,12,13\}$ to $\{8,10,12\}$ defined by $y=x-3$. Then, $R^{-1}$ is equal to

1. $\{(8,11),(10,13)\}$

2. $\{(11,18),(13,10)\}$

3. $\{(10,13),(8,11)\}$

4. None of these

Q. If $f\left(x\right)$ is an invertible function and $g\left(x\right)=2f\left(x\right)+5$, then the value of $g^{-1}\left(x\right)$, is

1. $2f^{-1}\left(x\right)-5$
2. $\frac{1}{2f^{-1}\left(x\right)+5}$
3. $\frac{1}{2}{f}^{-1}\left(x\right)+5$
4. $f^{-1}\left(\frac{x-5}{2}\right)$

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. $[0,\frac{\pi }{2}]$
2. $[-\frac{\pi }{4},\frac{\pi }{4}]$
3. $[-\frac{\pi }{2},\frac{\pi }{2}]$
4. $[0,\pi ]$

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

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

Q. If the equation ${\cos }^{-1}\left(\cos x\right)=\frac{n-x}{n}$, has seven roots for all $x\ge 0$, then $n\in$

1. $(0,2\pi )$
2. $(2\pi ,4\pi )$
3. $(4\pi ,6\pi )$
4. $(6\pi ,8\pi )$

Q. Number of real solution of the equation $x^3+3x^2-(x-6{)}^{\frac{1}{3}}+8+3x=0$ is

Q.

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

1. {(2, 1), (4, 2), (6, 3).....}

2. {(1, 2), (2, 4), (3, 6)....}

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

4. None of these

Q. Let $A=\{1,2,3\},B=\{1,3,5\}.$ A relation $R$ is defined from $A$ to $B$ as $R=\left\{\right(1,3),(1,5),(2,1\left)\right\}.$ Then $R^{-1}=$

1. $\left\{\right(1,2),(3,1),(1,3),(1,5\left)\right\}$
2. $\left\{\right(1,2),(3,1),(2,1\left)\right\}$
3. $\left\{\right(1,2),(5,1),(3,1\left)\right\}$
4. $\left\{\right(1,3),(1,5),(2,1\left)\right\}.$

Q. If $f\left(x\right)$ is an invertible function and $g\left(x\right)=2f\left(x\right)+5$, then the value of $g^{-1}\left(x\right)$, is

1. $2f^{-1}\left(x\right)-5$
2. $\frac{1}{2f^{-1}\left(x\right)+5}$
3. $\frac{1}{2}{f}^{-1}\left(x\right)+5$
4. $f^{-1}\left(\frac{x-5}{2}\right)$

Q. $\mathrm{If}p\left(x\right)=x^n\mathrm{and}q\left(x\right)=x^{\frac{1}{n}},\mathrm{where}n\mathrm{is}\mathrm{odd},\mathrm{then}p\left(x\right)\mathrm{and}q\left(x\right)\mathrm{are}\mathrm{inverse}\mathrm{of}\mathrm{each}\mathrm{other}.$

1. True
2. False

Q.

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

1. {(2, 1), (4, 2), (6, 3).....}

2. {(1, 2), (2, 4), (3, 6)....}

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

4. None of these