Find the inverse function of $2^{\sqrt{2-x}}-3$

Question

$2^{\sqrt{2-x}}-3$ How to find the inverse function this exponential function?

Answer 1

Basically, we have, $$y= 2^{\sqrt{2-x}} - 3$$ $$\Rightarrow y+3=2^{\sqrt{2-x}}$$ $$\Rightarrow \log_2(y+3) =\sqrt{2-x}$$ $$\Rightarrow (\log_2(y+3))^{2} = 2-x$$ $$ \Rightarrow x=2-(\log_2(y+3))^2.$$ Hence our inverse function is $$2-(\log_2(x+3))^2$$

Answer 2

$$ y = 2^{\sqrt{2-x}} - 3 \\ \iff y = e^{{\sqrt{2-x}} \ln(2)} - 3 \\ \iff \ln(y + 3) = {\sqrt{2-x}} \ln(2)\\ \Rightarrow \left(\frac{\ln(y + 3)}{\ln(2)}\right)^2 = 2-x \\ \Rightarrow x = 2 - \left(\frac{\ln(y + 3)}{\ln(2)}\right)^2 \\ $$