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Question

prove that function "f" defined as f:R(2)R1,f(x)=xx+2 is one - one and onto. Also find f1

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Solution

f(x)=xx+2
x+20

Domain of function =R{2}

x2

Therefore,

f(y)=yy+2

Let f(x)=f(y)

xx+2=yy+2

x(y+2)=y(x+2)

xy+2x=xy+2y

x=y

Hence the function is one-one.
Now, let y=f(x)

y=xx+2

y(x+2)=x

xy+2y=x

xxy=2y

x=2y1y=f(y).....(i)

1y0

y1

Hence range of the function is R{1}.

On replacing y with x in equation (i), we get inverse of f(x).

f1(x)=2x1x

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