Question

Given the function f(x) = 0.5(3)x, what is the value of f−1(7)?

This question has been asked before but the answer is unclear to me.

Answer 1

as you already know, to get the inverse of any expression, we start off by doing a quick switcharoo on the variables, and then solve for "y".


`\bf \stackrel{f(x)}{y}=0.5(3)^x\qquad \stackrel{inverse}{\boxed{x}=0.5(3)^{\boxed{y}}} \\\\\\ log(x)=log[0.5(3)^y]\implies log(x)=log(0.5)+log[(3)^y] \\\\\\ log(x)-log(0.5)=log[(3)^y]\implies log(x)-log(0.5)=y\cdot log(3) \\\\\\ \cfrac{log(x)-log(0.5)}{log(3)}=\stackrel{f^(-1)(x)}{y}\\\\ -------------------------------\\\\ f^(-1)(7)=\cfrac{log(7)-log(0.5)}{log(3)}\implies f^(-1)(7)\approx 2.402173502732879697`