Prove that the classes of decidable and recognizable languages are closed under application of inverses

of computable functions, that is$,$ if f : $\backslash$Sigma $$

$1->\backslash$Sigma

$$

$2$

is a computable function and L $\backslash$Sigma

$$

$2$

is a decidable

$($resp$.$ recognizable$)$ language, then f

$1$

$($L$)=\{$x in $\backslash$Sigma

$$

$1$

$|$ f$($x$)$ in L$\}$ is a decidable $($resp$.$ recognizable$)$

language.