How to Find the Inverse of a Function

Q. The inverse function of $f\left(x\right)=\frac{8^{2x}-8^{-2x}}{8^{2x}+8^{-2x}},x\in (-1,1)$, is

1. $\frac{1}{4}\left({\log }_{8}e\right){\log }_e\left(\frac{1-x}{1+x}\right)$
2. $\frac{1}{4}\left({\log }_{8}e\right){\log }_e\left(\frac{1+x}{1-x}\right)$
3. $\frac{1}{4}{\log }_e\left(\frac{1+x}{1-x}\right)$
4. $\frac{1}{4}{\log }_e\left(\frac{1-x}{1+x}\right)$

Q.

If ${\sin }^{-1}\left(x\right)+{\sin }^{-1}\left(y\right)=\frac{2\pi }{3}$, then ${\cos }^{-1}\left(x\right)+{\cos }^{-1}\left(y\right)=$

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

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

3. $\frac{\mathrm{\pi }}{6}$

4. $\mathrm{\pi }$

Q. Let $A=\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right].$ Then the number of $3\times 3$ matrices $B$ with entries from the set $\{1,2,3,4,5\}$ and satisfying $AB=BA$ is

Q.

The domain of the function $f\left(x\right)={\log }_2\left(-{\log }_{\frac{1}{2}}\left(1+\frac{1}{x^{{\displaystyle \frac{1}{4}}}}\right)-1\right)$ is

1. $\left(0,1\right)$

2. $(0,1]$

3. $[1,\infty )$

4. $\left(1,\infty \right)$

Q.

Evaluate ${\sin }^4\left(\frac{\pi }{8}\right)+{\sin }^4\left(\frac{3\pi }{8}\right)+{\sin }^4\left(\frac{5\pi }{8}\right)+{\sin }^4\left(\frac{7\pi }{8}\right)$:

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

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

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

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

Q.

If f and g two functions such that they are one-one then g o f is:

1. a bijective function

2. a many-one function

3. a many-one and onto function

4. a one-one function

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

Q.

If : A→B and g : B →C be two functions. Then, composition of f and g, gof : A→C is defined as

1. (gof)(x) = f(x), for all x ∈ A

2. (gof)(x) = g(f(x)), for all x ∈ A

3. (gof)(x) = f(g(x)), for all x ∈ A

4. (gof)(x) = g(x), for all x ∈ A

Q.

Solve the equations: $2x^4+x^3-6x^2+x+2=0.$

Q. For real $x$, the greatest value of $\frac{x^2+2x+4}{2x^2+4x+9}$ is

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

Q.

Draw the graph of the function y = $a^x$ when 0 < a < 1.

1. 

2. 

3. 

4. 

Q. Which among the following function is continuous everywhere in its domain but has at least one point where it is not differentiable

1. $f\left(x\right)=x^{\frac{1}{3}}$
2. $f\left(x\right)=\frac{|x|}{x}$
3. $f\left(x\right)=e^{-x}$
4. $f\left(x\right)=\tan x$

Q.

The mirror image of the curve $Arg\left(\frac{z+i}{z-1}\right)=\frac{pi}{4}$across the real line $x-y=0$

Q.

Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f⁻¹ and show that (f⁻¹)⁻¹ = f.

Q. $If3+\frac{3+d}{4}+\frac{3+2d}{4^2}+....to\infty =8.thend=$

1. 9
2. 10
3. 12
4. 15

Q. What is proof of the derivative of sec inverse (x)

Q. Let $f\left(x\right)=\frac{1+x}{1-x}.$ If $A$ is matrix for which $A^3=O,$ then $f\left(A\right)$ is

1. $I+A+A^2$
2. $I+2A+2A^2$
3. $I-A-A^2$
4. $O$

Q.

The function $f\left(x\right)=(x/3)+(3/x),x\ne 0$decreases in the interval.

1. $(-\infty ,3)$

2. $(-3,3)$

3. $(3,\infty )$

4. $(-9,9)$

Q.

Let f:[0, infinity) to R be a function defined by f(x)= 9x² +6x-5. Proove that f is not invertible.Modify , only the codomain of f to make f invertible and then find it's inverse .Also , find f^{​​​​​​-1}(43)

Q.

The area bounded by $y=2-|2-x|;y=\frac{3}{|x|}is\frac{k-3ln3}{2},$ then k = ___

Q. Let $f:[\frac{\pi }{3},\frac{2\pi }{3}]\to [3,4]$ be a function defined by $f\left(x\right)=\sqrt{3}\sin x-\cos x+2.$ Then $f^{-1}\left(x\right)$ is given by

1. ${\cos }^{-1}\left(\frac{x-2}{2}\right)+\frac{2\pi }{3}$
2. ${\sin }^{-1}\left(\frac{x-2}{2}\right)-\frac{\pi }{6}$
3. $f^{-1}$ does not exist
4. ${\sin }^{-1}\left(\frac{x-2}{2}\right)+\frac{\pi }{6}$

Q. 2. Let f be defined by f(x)=x-4 and g be defined by g(x)={x-16/x+4 when x is not equal to -4 Or @ when x=-4 } Find @ such that f(x)=g(x) for all x.

Q.

The value of ${\int }_{-\mathrm{\pi }}^{\mathrm{\pi }}\left({\sin }^{3}x\right)\left({\cos }^{2}x\right)dx$ is

1. $1$

2. $2$

3. $3$

4. $0$

Q.

Show that $f:[-1,1]\to R$, given by $f\left(x\right)=\frac{x}{(x+2)},x\ne -2$, is one-one. Find the inverse of the function $f:[-1,1]\to$ Range f.

Q. Let S = { a , b , c } and T = {1, 2, 3}. Find F −1 of the following functions F from S to T , if it exists. (i) F = {( a , 3), ( b , 2), ( c , 1)} (ii) F = {( a , 2), ( b , 1), ( c , 1)}

Q. Let $f:R\to R$ be defined by $f\left(x\right)=2x+6$ which is a bijective mapping then $f^{-1}\left(x\right)$ is given by

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

Q. Let $f:R\to R,g:R\to R$, be two functions, such that f(x) =2x – 3, g (x) = $x^3$ + 5.
The function $(fog{)}^{-1}$ (x) is equal to

1. ${(x-\frac{7}{2})}^{\frac{1}{3}}$
2. ${\left(\frac{x-7}{2}\right)}^{\frac{1}{3}}$
3. ${\left(\frac{x+7}{2}\right)}^{\frac{1}{3}}$
4. ${\left(\frac{x-2}{7}\right)}^{\frac{1}{3}}$

Q. The inverse of the function {(1, 2), (2, 3), (3, 1)} defined from set A = {1, 2, 3} to itself is

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

Q. Let be a function defined as . The inverse of f is map g : Range (A) (B) (C) (D)

Q.

Let M(n) be the largest integer in m such that ${}^m{C}_{n-1}{>}^{m-1}{C}_n,$ then the value of $\underset{n\to \infty }{lim}$ $\frac{M\left(n\right)}{n}$ is

1. $\frac{3-\sqrt{5}}{2}$

2. $\frac{3+\sqrt{5}}{2}$

3. $\frac{\sqrt{5}-1}{2}$

4. $\frac{\sqrt{5}+1}{2}$