Expert: Sherman D. - 9/11/2005

Question
find f^-1 and verify that (f o f^-1)(x)= (f^-1 o f)(x)=x

1) f(x)= 1/x^2, x > 0
2) f(x)= 2x+1/x+3

Answer
1.)
f(x) = 1/(x^2)
y = 1/(x^2)
x = 1/(y^2)
y^2 = 1/x
y = sqrt(1/x)

ANS : f(x)' = sqrt(1/x)

Proofs
f(f(x)')
f(sqrt(1/x)) = 1/((sqrt(1/x))^2)
f(sqrt(1/x)) = 1/(1/x)
f(sqrt(1/x)) = (1/1)/(1/x)
f(sqrt(1/x)) = (1/1)*(x/1)
f(sqrt(1/x)) = x

f(f(x))'
f(1/(x^2)) = sqrt(1/(1/(x^2)))
f(1/(x^2)) = sqrt((1/1)/(1/(x^2))))
f(1/(x^2)) = sqrt((1/1)*((x^2)/1)))
f(1/(x^2)) = sqrt(x^2)
f(1/(x^2)) = x

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2.)
f(x) = (2x + 1)/(x + 3)
y = (2x + 1)/(x + 3)
x = (2y + 1)/(y + 3)
xy + 3x = 2y + 1
xy - 2y = -3x + 1
y(x - 2) = -3x + 1
y = (-3x + 1)/(x - 2)

ANS : f(x)' = (-3x + 1)/(x - 2)

Proofs
f(f(x))'
f((2x + 1)/(x + 3))' = (-3((2x + 1)/(x + 3)) + 1)/(((2x + 1)/(x + 3)) - 2)
f((2x + 1)/(x + 3))' = (((-6x - 3)/(x + 3)) + 1)/(((2x + 1)/(x + 3)) - 2)
f((2x + 1)/(x + 3))' = (((-6x - 3) + (x + 3))/(x + 3))/(((2x + 1) - (2x + 6))/(x + 3))
f((2x + 1)/(x + 3))' = ((-6x - 3 + x + 3)/(x + 3))/((2x + 1 - 2x - 6)/(x + 3))
f((2x + 1)/(x + 3))' = ((-5x)/(x + 3))/(-5/(x + 3))
f((2x + 1)/(x + 3))' = ((-5x)/(x + 3))*((x + 3)/(-5))
f((2x + 1)/(x + 3))' = (-5x(x + 3))/(-5(x + 3))

the (x + 3)s cancel out

f((2x + 1)/(x + 3))' = (-5x)/(-5)

the -5s cancel out

f((2x + 1)/(x + 3))' = x

f(f(x)')
f((-3x + 1)/(x - 2)) = (2((-3x + 1)/(x - 2)) + 1)/(((-3x + 1)/(x - 2)) + 3)
f((-3x + 1)/(x - 2)) = (((-6x + 2)/(x - 2)) + 1)/(((-3x + 1) + (3x - 6))/(x - 2))
f((-3x + 1)/(x - 2)) = (((-6x + 2) + (x - 2))/(x - 2))/((-3x + 1 + 3x - 6)/(x - 2))
f((-3x + 1)/(x - 2)) = ((-6x + 2 + x - 2)/(x - 2))/((-5)/(x - 2))
f((-3x + 1)/(x - 2)) = ((-5x)/(x - 2))/((-5)/(x - 2))
f((-3x + 1)/(x - 2)) = ((-5x)/(x - 2))*((x - 2)/(-5))
f((-3x + 1)/(x - 2)) = (-5x(x - 2))/(-5(x - 2))
f((-3x + 1)/(x - 2)) = x