Mathematics - Functions Question with Solution | TestHub

$\begin{array}{rl}& f(n)=\{\begin{array}{ll}(n+1)/2& \text{ if }n\text{ is odd }\\ (4-n)/2& \text{ if }n\text{ is even }\end{array}\\ & f(n)=\{(1,1),(2,1),(3,2),(4,0),(5,3),(6,-1),\dots \}\end{array}$ $\therefore \mathrm{f}(\mathrm{n})$ is many one and onto function $\begin{array}{rl}& \mathrm{g}(\mathrm{n})=\{\begin{array}{cc}3+2\mathrm{n}& \text{ if }\mathrm{n}\ge 0\\ -2\mathrm{n}& \text{ if }\mathrm{n}<0\end{array}\\ & \mathrm{g}(\mathrm{n})=\{(-3,6),(-2,4),(-1,2),(0,3),(1,5),(2,7),(3,9),(4,15),\dots \}\end{array}$ $\therefore \mathrm{g}(\mathrm{n})$ is one-one and into function $\mathrm{f}(\mathrm{g}(\mathrm{n}))=2+\mathrm{n},\mathrm{n}\in \mathrm{N}$ fog is one-one and into $g(f(n))=\{\begin{array}{ll}4+n& \text{ if }n\text{ is odd natural number }\\ 7-n& \text{ if }n=2,4\\ n-4& \text{ if }n\text{ is even natural number and }n\ge 6\end{array}$ $g(f(2))=g(f(1))=5$ $\therefore$ gof is many one and into