# Math Help - help me in this function question

1. ## help me in this function question

if the function is
f:[1,INFINITY]->[1,infinity] is defined by
f(x)=2^(x(x-1)) then
f inverse x=?

2. $y= 2^{x(x-1)}$

swap x and y and solve for y

$x= 2^{y(y-1)}$

$y(y-1) = \log_2 (x)$

$y^2-y = \log_2 (x)$

now complete the square

$\left(y^2-y +\frac{1}{4}\right) -\frac{1}{4}= \log_2 (x)$

$\left(y-\frac{1}{2}\right)^2 -\frac{1}{4}= \log_2 (x)$

$\left(y-\frac{1}{2}\right)^2 = \log_2 (x)+\frac{1}{4}$

$y-\frac{1}{2} = \sqrt{\left(\log_2 (x)+\frac{1}{4}\right)}$

$y = \sqrt{\left(\log_2 (x)+\frac{1}{4}\right)}+\frac{1}{2}$

therefore

$f^{-1}(x) = \sqrt{\left(\log_2 (x)+\frac{1}{4}\right)}+\frac{1}{2}$