Question

Let $f:D\to R$, where $D$ is the domain of $f$. Find the inverse of $f$, if it exists
$f\left(x\right)={1-2}^{-x}$
$f\left(x\right)={(4-{(x-7)}^3)}^{1/5}$
$f\left(x\right)=\ell n(x+\sqrt{{1+x}^2})$

Answer 1

$\left(1\right)$ $f\left(x\right)=1-2^{-x}$

$y=1-2^{-x}$

for $f^{-1}\left(x\right)$ , Put $x=y$ and $y=x$

$x=1-2^y$

$-x+1=2^y$

$-y={\log }_2(1-x)$

$y={\log }_2\left(1/\right(1-x\left)\right)=f^{-1}\left(x\right)$

$\left(2\right)$ $y=(4-(x-7{)}^3{)}^{1/5}$

interchange $x$ and $y$

$x=(4-(y-7{)}^3{)}^{1/5}$

$(y-7{)}^3=4-x^5$

$y=7+(4-x^5{)}^{1/3}=f^{-1}(x)$

$\left(3\right)$ $y=\ln (x+\sqrt{(1+x)}$

interchange $x$ and $y$

$x=\ln (y+\sqrt{(1+y)}$

$e^x=y+\sqrt{(1+y^2}$

$(e^x-y{)}^2=1+y^2$

$y=\frac{e^{2x}-1}{2e^x}=f^{-1}\left(x\right)$