This page offers a step-by-step solution to the specific question **NCERT Class 12th Mathematics - Relations and Functions | nbsp if f x nbsp show that fof x x for a Answer ** from NCERT Class 12th Mathematics, Chapter Relations and Functions.

Question 4

If *f(x)* = _{}, show that *fof*(x) = x, for all x ≠ 2/3. What is the inverse of* f *?

Answer

It is given that.

Hence, the given function *f* is invertible and the inverse of *f* is *f* itself.

- Q:- Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive. - Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:

(i) Relation R in the set A = {1, 2, 3,13, 14} defined as

R = {(x, y): 3x − y = 0}

(ii) Relation R in the set N of natural numbers defined as

R = {(x, y): y = x + 5 and x < 4}

(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as

R = {(x, y): y is divisible by x}

(iv) Relation R in the set Z of all integers defined as

R = {(x, y): x − y is as integer}

(v) Relation R in the set A of human beings in a town at a particular time given by

(a) R = {(x, y): x and y work at the same place}

(b) R = {(x, y): x and y live in the same locality}

(c) R = {(x, y): x is exactly 7 cm taller than y}

(d) R = {(x, y): x is wife of y}

(e) R = {(x, y): x is father of y} - Q:- Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
- Q:- Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as

R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. - Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b
^{2}} is neither reflexive nor symmetric nor transitive. - Q:-
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i)

*f*:**R → R**defined by*f(x)*= 3 – 4x(ii)

*f*:**R → R**defined by*f(x)*= 1 + x^{2 } - Q:- Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
- Q:-
Show that the Modulus Function

*f*: R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative. - Q:-
Prove that the Greatest Integer Function

*f*: R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. - Q:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.

- Q:-
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5.

- Q:-
Determine order and degree(if defined) of differential equation \begin{align} \frac{d^4y}{dx^4}\;+\;\sin(y^m)\;=0\end{align}

- Q:-
Represent graphically a displacement of 40 km, 30° east of north.

- Q:-
If a line makes angles 90°, 135°, 45° with

*x*,*y*and*z*-axes respectively, find its direction cosines. - Q:-
Maximise Z = 3

*x*+ 4*y*Subject to the constraints:

*x*+*y*≤ 4,*x*≥ 0,*y*≥ 0 - Q:-
Find the area of the region bounded by the curve

*y*^{2}=*x*and the lines*x*= 1,*x*= 4 and the*x*-axis. - Q:- Evaluate the determinants

\begin{vmatrix} \mathbf{2} & \mathbf{4} \\ \mathbf{-5} & \mathbf{-1} \end{vmatrix} - Q:- Find the rate of change of the area of a circle with respect to its radius r when

(a) r = 3 cm

(b) r = 4 cm - Q:-
Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E).

- Q:- Integrals sin 2x

- Q:- Find the principal value of \begin{align} cosec^{-1}\left({2}\right)\end{align}
- Q:-
Determine order and degree(if defined) of differential equation y

^{m}+ 2y^{n}+ y' =0 - Q:- Find the principal value of \begin{align} cot^{-1}\left(\sqrt3\right)\end{align}
- Q:-
An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

- Q:- Find the principal value of \begin{align} cosec^{-1}\left({-\sqrt2}\right)\end{align}
- Q:-
\begin{align} y = xsinx:xy{'}=y +x\sqrt{x^2 -y^2}(x\neq0\; and\; x>y\; or\; x<-y)\end{align}

- Q:- If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
- Q:- Find the principal value of \begin{align} tan^{-1} (1) + cos^{-1}\left(-\frac{1}{2}\right) + sin^{-1}\left(-\frac{1}{2}\right)\end{align}
- Q:- Integrals e
^{2x} - Q:- Find the principal value of \begin{align} tan^{-1}\left(-\sqrt3\right)\end{align}

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