Question

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$. Define
$f: S \rightarrow S$ as $f(n)=\left\{\begin{array}{cl}2 n, & \text { if } n=1,2,3,4,5 \\ 2 n-11 & \text { if } n=6,7,8,9,10\end{array}\right.$
Let $g : S \rightarrow S$ be a function such that $f o g(n)=\begin{cases}n+1 & \text {, if } n \text { is odd } \\ n-1 & \text {, if } n \text { is even }\end{cases}$, then $g (10)(( g (1)+ g (2)+ g (3)+ g (4)+ g (5))$ is equal to:

Answer 1

$f ^{-1}( n )=\begin{cases}\frac{ n }{2} & ; n =2,4,6,8,10 \\ \frac{ n +11}{2} & ; \quad n =1,3,5,7,9\end{cases}$
$f ( g ( n ))= \begin{cases} n +1 & ; n \in \text { odd } \\ n -1 & ; \quad n \in \text { even }\end{cases}$
$g ( n )= \begin{cases} f ^{-1}( n +1) & ; n \in \text { odd } \\ f ^{-1}( n -1) & ; \quad n \in \text { even }\end{cases}$
$g ( n )= \begin{cases}\frac{ n +1}{2} ; & n \in \text { odd } \\ \frac{ n +10}{2} ; & n \in \text { even }\end{cases}$
$g (10) \cdot\left[ g (1)+ g (2)+ g (3)+ g (4)+ g (5)\right]$
$=10 \cdot\left[1+6+2+7+3]=190\right]$