Question

Find the range of \[x=\frac{ 2^y-2^{-y} }{ 2^y + 2^{-y} }\]

Answer 1

let's rewrite it as
\[ \large y=\frac{2^x-2^{-x}}{2^x+2^{-x}}=\frac{2^{2x}-1}{2^{2x}+1} \]

Answer 2

\[ \large y2^{2x}+y=2^{2x}-1 \]
\[ \large 2^{2x}(y-1)=-y-1 \]
\[ \large 2^{2x}=\frac{-y-1}{y-1}=\frac{y+1}{1-y} \]
\[ \large 2x=\log_2\frac{y+1}{1-y} \]
\[ \large x=\frac{1}{2}[\log_2(y+1)-\log_2(1-y)] \]

Answer 3

i guess u can finish from this.