Question

The function (f) is given by -11) -8" alt="f(x)=e^(x-11) -8" align="absmiddle" class="latex-formula">
a) Find $f^-1 (x)$

b) Write down the domain of $f^-1 (x)$

Answer 1

$f(x)=e^{x-11}-8\to y=e^{x-11}-8\ \ \ \ |add\ 8\ to\ both\ sides\\\\e^{x-11}=y+8\\\\\ln e^{x-11}=\ln(y+8)\iff x-11=\ln(y+8)\ \ \ \ |add\ 11\ to\ both\ sides\\\\x=\ln(y+8)+11\\\\a)\ \boxed{f^{-1}(x)=\ln(x+8)+11}\\\\b)\ The\ domain\ of\ f^{-1}(x):\\\\x+8 > 0\to \boxed{x > -8\to x\in(-8;\ \infty)}$