• 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.,)

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 = IRange f

∴ f - 1 = g

⇒

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

• Q:- Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = { (a,b) ; |a - b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
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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.

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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:- 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.
• ## Recently Viewed Questions of Class 12th mathematics

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

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• 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
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(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}
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(i) f : N → N given by f(x) = x2

(ii) f : Z → Z given by f(x) = x2

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(iv) f : N → N given by f(x) = x3

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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