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Question

Which of the following functions is inverse of itself?


A

f(x)=1-x1+x

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B

f(x)=3logx

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C

f(x)=3x(x+1)

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D

None of these

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Solution

The correct option is A

f(x)=1-x1+x


Explanation for the correct option:

Finding the inverse of the function

Given f(x)=1-x1+x

y=f(x)f-1(y)=x

Therefore,

y=1-x1+xy(1+x)=1-xy+yx=1-xyx+x=1-yx(y+1)=1-yx=1-y1+y

Hence, functions 1-x1+x is inverse of itself.

Explanation for the incorrect options:

Finding the inverse of the function of 3logx

For option (b)

Given f(x)=3logx

y=f(x)f-1(y)=x

Therefore,

y=3logxlogx=y3x=ey3

Hence, function 3logx is not the inverse of itself.

Finding the inverse of the function of 3x(x+1)

For option (c)

Given f(x)=3x(x+1)

y=f(x)

y=3xx+1

3x2+3x-y=0

x=-3±32-4×3×-y2×3

x=-3±9+12y6

Replace x by y and y by x

y=-3±9+12x6

Therefore y=-3±9+12x6

f-1x=-3±9+12x6

Hence, function 3x(x+1) is not the inverse of itself.

Hence, functions 1-x1+x is inverse of itself.

Therefore, the correct option is option (A).


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