Q.

Which of the following functions is an injective (one-one) function in its respective domain?

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a

f(x)=2x+sin⁡3x

b

f(x)=x⋅[x], (where [.] denotes the G.I.F)

c

f(x)=2x−14x+1

d

fx =2x+14x-1

answer is D.

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Detailed Solution

1 If fx=2x +sin3x       ⇒ f'x =2 +3 cos3x =0      ⇒    cos 3x=-23 ,which is defined         and  f''x =- 9 sin3x = -35<0    so fx  has maximum value exists in its domain , so it is not one-one2  fx = x.x is not one-one since f0.1=0.1  ×0 =0 and f0.2=0                                                 3   fx=2x-14x+1 ⇒ fx is continuous function and f0=0 and fx →0 again when x→∞      since y =12x-14x1+14x →0 as x→∞         hence fx  has a atleat one maximum exists , so it is not one-one4  fx =2x+14x-1 =2x+12x-12x+1 =12x-1       ⇒ f'x = -2x log22x-12<0 ⇒fx  is decreasing function   so it is one-one
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Which of the following functions is an injective (one-one) function in its respective domain?