$$ f(x) = \begin{cases} 0 &\text{ if } |u| \le 1 \\ |u|-1 &\text{ if } |u| \gt 1 \end{cases} $$ $$ g(x) = \begin{cases} u^2 &\text{ if } |u| \leqslant 1 \\ 2|u|-1 &\text{ if } |u| > 1 \end{cases}$$
$$f^*(s) = \sup \left( \langle s,x\rangle -f(x)\right)$$
I am studying this for the first time. To learn from easier examples, Please help me to find the conjugate functions of $f(x),g(x)$
Thank you.