Question

Range of the function \[Z=\frac{ 2^y - 2^{-y} }{ 2^y + 2^{-y} }\]

Answer 1

easier to see if you multiply top and bottom by \(2^x\)

Answer 2

you see more or less instantly that \(y<1\) for all \(x\)

Answer 3

then let \(x\to -\infty\) and see that the limit is \(-1\)

Answer 4

\[Z=\frac{ 2^y - 2^{-y} }{ 2^y + 2^{-y} }\]

Answer 5

so Z is our variable

Answer 6

Yup..)

Answer 7

\[Z=\frac{ 2^{2y}+1 }{ 2^y }\times \frac{ 2^{2y}+1 }{ 2^{y} }=\frac{ (2^{2y}+1)^2 }{ 2^{2y} }\]
the inverse function is this
\[f^{-1}Z=\frac{ (2^{2Z}+1)^2 }{ 2^{2Z} }\] above and if we can tell its domain then we can swap it to be the range\[2^{2Z}=0\]
so \[Z \in \mathbb{R}\]

Answer 8

hence \[y \in \mathbb{R}\]