Question:
Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.

(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,)

Answer:

f: [ - 1, 1] → R is given as

Let f(x) = f(y).

∴ f is a one-one function.

It is clear that f: [ - 1, 1] → Range f is onto.

∴ f: [ - 1, 1]→ Range f is one-one and onto and therefore, the inverse of the function:

f: [ - 1, 1] → Range f exists.

Let g: Range f → [ - 1, 1] be the inverse of f.

Let y be an arbitrary element of range f.

Since f: [ - 1, 1] → Range f is onto, we have:

Now, let us define g: Range f → [ - 1, 1] as

∴gof =I_{[-1, 1]}and fog = I_{Range f}

∴ f - 1 = g

⇒

1
0
0
Report?

Report a problem on Specifications:

Email id:

Comments

Next Question >

SELECT ex_no,question,question_no,id,chapter FROM question_mgmt as q WHERE courseId='3' AND subId='6' AND ex_no!=0 AND status=1 and id!=1557 ORDER BY views desc, last_viewed_on desc limit 0,10

Popular questions of class 12 Mathematics

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 ≤ b²} is neither reflexive nor symmetric nor transitive.

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 + x²

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.

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.

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.

SELECT ex_no,question,question_no,id,chapter FROM question_mgmt as q WHERE courseId='3' AND subId='6' AND ex_no!=0 AND status=1 and id!=1557 ORDER BY last_viewed_on desc limit 0,10

Recently Viewed Questions of Class 12 Mathematics

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: sin 2x – 4e^3x

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Q: If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?

The order of the differential equation

\begin{align}2x^2\frac{d^2y}{dx^2}\;- \;3\frac{dy}{dx}\;+ y=\;0\end{align}

is (A) 2 (B) 1 (C) 0 (D) not defined

Determine order and degree(if defined) of differential y^' + y =e^x

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

Q: Find the principal value of \begin{align} cosec^{-1}\left({-\sqrt2}\right)\end{align}

Show that the Signum Function f : R → R, given by

is neither one-one nor onto.

Comments

Answered by Ekta Mehta
4 months ago

Taking Screenshots on your Samsung Galaxy M31s is very easy and quick.

1
0
0
Report?

Report a problem on Specifications:

Email id:

Comments

Answered by Ekta Mehta
4 months ago

Taking Screenshots on your Samsung Galaxy M31s is very easy and quick.

1
0
0
Report?

Report a problem on Specifications:

Email id:

Comments

Answered by Ekta Mehta
4 months ago

Taking Screenshots on your Samsung Galaxy M31s is very easy and quick.

1
0
0
Report?

Report a problem on Specifications:

Email id:

Comments

Question:
Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.

(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,)

< Previous Question

Next Question >

Popular questions of class 12 Mathematics

Recently Viewed Questions of Class 12 Mathematics

Comments

Comment(s) on this Question

Important links

Quick Links

Resources

Support

Question: Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f. (Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,)

< Previous Question

Next Question >

Popular questions of class 12 Mathematics

Recently Viewed Questions of Class 12 Mathematics

Comments

Comment(s) on this Question

Important links

Quick Links

Resources

Support

Question:
Show that f : [–1, 1] → R, given by is one-one. Find the inverse of the function f : [–1, 1] → Range f.

(Hint: For y ∈ Range f, y =, for some x in [ - 1, 1], i.e.,)