Question #38472

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Answer 1 · 2017-11-14T17:57:10

#f^-1(x)=(1+sqrt(1-4x^2))/(2x)#

#f^-1(x)=(1-sqrt(1-4x^2))/(2x)#

To find the inverse we need to express #f(x)# as a function of y:

Multiply both sides by #(x^2+1)#

#y(x^2+1)=x#

#x=yx^2+y#

Subtract #yx^2+y#

#x-yx^2-y=0#

#yx^2-x+y=0#

Solve with the quadratic formula:

#x= (1+-sqrt(1-(4y^2)))/(2y)#

#x=(1+sqrt(1-4y^2))/(2y)#

#x=(1-sqrt(1-4y^2))/(2y)#

Substituting #y=x#

#y=(1+sqrt(1-4x^2))/(2x)#

#y=(1-sqrt(1-4x^2))/(2x)#

or:

#f^-1(x)=(1+sqrt(1-4x^2))/(2x)#

#f^-1(x)=(1-sqrt(1-4x^2))/(2x)#