How to Find the Inverse of a Function

Q.

Find the domain and the range of the real function, $f\left(x\right)=\frac{3}{(2-x^2)}$.

Q.

Consider the equation $x^2+2x-n=0$, where n $\in$ [5, 100]. Total number of different values of 'n' so that the given equation has integral roots is

1. 4

2. 6

3. 8

4. 3

Q. If f, g:R-R are two functions defined as f(X) =|X|+X and g(X)=|X|-X , for every X belong to R then, find fog and god.

Q. Let $f\left(x\right)=x^2-x+1$, $x\ge \frac{1}{2}$ then the solution of the equation $f^{-1}\left(x\right)=x$ is

1. x = 1
2. x = 2
3. $x=\frac{1}{2}$
4. None of these

Q. Find the value of $(a^2+\sqrt{a^2-1}{)}^4+(a^2-\sqrt{a^2-1}{)}^4$.

Q. If f, g:R→R be two functions defined as f(x)=∣x∣+x and g(x)=∣x∣−x for all x∈R. Then, find f∘g and g∘f.

Q. If a function f : R → R be defined by
$f\left(x\right)=\left\{\begin{array}{cc}3x-2,& x<0;\\ 1,& x=0;\\ 4x+1,& x>0.\end{array}\right.$
find: f(1), f(−1), f(0) and f(2).

Q. The period of the function if g(x)+g(x+4)=g(x+2)+g(x+6) is (A)4 (B)6 (C)8 (D)1

Q.

Find the value of 1 + $\omega$ + ${\omega }^2$ + ${\omega }^3$

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Q. f, g:R->R be two functions given by f(x) =x+|x| and g(x) =-x+|x| for all values of x belongs to R find fog and gof

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

Q. Write the value of $\underset{x\to 1^{-}}{\lim }x-\left[x\right].$

Q. if, F(x)=(1+1/x)^-1 and g(x)=(x+1/x)^-1 then, what is fog ?

Q. Let $f:R\to R$ be defined as f(x) = 10x + 7. The function $g:R\to R$ such that gof = fog =$I_R$. Then g(2017) =

Q.

If A + B + C = $\pi$, Prove that

cos 4A + cos 4B + cos 4C = -1 + 4 cos 2A cos 2B cos 2C.

Q. 57 let g(x)=1+x-[x] and f(x)={-1, x<0;0, x=0;1, x>0, then for all x, f(g(x)) is equal to

Q.

If A + B + C =$\pi$. Prove that

cos 2A + cos 2B + cos 2C = -1 -4 cos A cos B cos C.

Q. [x]+[2x]+[4x]+[8x]+[16x]+[32x]=12345. What is the real value of x for which the above equation holds true.

Q. If f, g:R→R be two functions defined as f(x)=∣x∣+x and g(x)=∣x∣−x for all x∈R. Then, find f∘g and g∘f.

Q. If \(f : R \rightarrow R\) be a mapping defined by $f\left(x\right)=x^3+5$, then $f^{-1}x$ is equal to

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

Q. Suppose f(x) = $(x+1{)}^2$ for $\ge$ -1. If g(x) is the function whose graph is reflection of the graph of f(x) with respect to line y = x, then g(x) equals

1. $-\sqrt{x}-1,x\ge 0$
2. $\frac{1}{(x+1{)}^2},x>-1$
3. $\sqrt{x+1},x\ge -1$
4. $\sqrt{x}-1,x\ge 0$

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

Q.

Graph of f(x) is given. Draw the graph of f${}^{-1}$(x)

1. 

2. 

3. 

4. 

Q. If f(x) = (x − a)² (x − b)², find f(a + b).