• NCERT Chapter
Question 10

# Let f : X → Y be an invertible function. Show that f has unique inverse.(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = 1Y(y) = fog2(y). Use one-one ness of f).

Let fX  → Y be an invertible function.

Also, suppose f has two inverses (say g1 and g2).

Then, for all y ∈Y, we have: Hence, f has a unique inverse.

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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.

• 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.

• 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:- 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.
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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.

• Q:-

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

• Q:-

Find gof and fog, if

(i) f(x) = | x | and g(x) = | 5x – 2 |
(ii) f(x) = 8x3 and g(x) = x1/3 .

• Q:-

Let f: X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f–1)–1 = f.

• Q:-

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