- NCERT Chapter

Question 6

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

∴*g*o*f* =I_{[-1, 1]}and *f*o*g* = I_{Range f}

∴ *f* - 1 = *g*

⇒

- 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:- Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by

(i) R = { (a,b) : |a - b| is a multiple of 4}

(ii) R = {(a,b):a = b} is an equivalence relation.

Find the set of all elements related to 1 in each case. - 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:-
Check the injectivity and surjectivity of the following functions:

(i)

*f*:**N → N**given by*f(x*) = x^{2}(ii)

*f*:**Z → Z**given by*f(x)*= x^{2}(iii)

*f*:**R → R**given by*f(x)*= x^{2}(iv)

*f*:**N → N**given by*f(x)*= x^{3}(v)

*f*:**Z → Z**given by*f(x)*= x^{3 } - 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:- . Is f one-one and onto? Justify your answer.

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Let A = R – {3} and B = R – {1}. Consider the function *f* : A → B defined by

. Is f one-one and onto? Justify your answer.

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.

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:- Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.

(A) R is reflexive and symmetric but not transitive.

(B) R is reflexive and transitive but not symmetric.

(C) R is symmetric and transitive but not reflexive.

(D) R is an equivalence relation. - Q:- If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
- Q:- . Is f one-one and onto? Justify your answer.

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Let A = R – {3} and B = R – {1}. Consider the function *f* : A → B defined by

. Is f one-one and onto? Justify your answer.

If a line has the direction ratios −18, 12, −4, then what are its direction cosines?

Find gof and fog, if

(i)* f(x)* = | x | and *g(x)* = | 5x – 2 |

(ii) *f(x)* = 8x3 and *g(x)* = x^{1/3} .

Let *f*: X → Y be an invertible function. Show that the inverse of *f ^{–1}* is f, i.e.,

If a line makes angles 90°, 135°, 45° with *x*, *y* and *z*-axes respectively, find its direction cosines.

(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}

Check the injectivity and surjectivity of the following functions:

(i)* f* : **N → N** given by* f(x*) = x^{2}

(ii)* f* : **Z → Z** given by *f(x)* = x^{2}

(iii)* f* : **R → R** given by* f(x)* = x^{2}

(iv)* f *: **N → N** given by *f(x)* = x^{3}

(v)* f* : **Z → Z** given by *f(x)* = x^{3 }

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 }

VIDIT
2019-05-03 14:20:46

WHERE IS THE RANGE OF THE FUNCTION