Question

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

Answer 1

Answer: 190

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