What is the inverse of $f(x) = (x+1)/(2x+1)$ ?

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Answer 1 · 2015-06-06T16:56:30

$f(x) = (x+1)/(2x+1) = (2(x+1))/(2(2x+1))$

$=((2x+1)+1)/(2(2x+1))$

$=1/2+1/(4x+2)$

Subtract $1/2$ from both sides to get:

$1/(4x+2) = f(x)-1/2$

Take the reciprocal of both sides to get:

$4x+2 = 1/(f(x)-1/2)$

Subtract $2$ from both sides to get:

$4x = 1/(f(x)-1/2) - 2$

Divide both sides by $4$ to get:

$x=1/(4(f(x)-1/2)) - 2$

$=1/(4f(x)-2) - 2$

So

$f^-1(y) = 1/(4y-2) - 2$

What is the inverse of f(x) = (x+1)/(2x+1)f(x) = (x+1)/(2x+1)?