Let \(k\) be a real number, \(k \ne \pm 1\). The function \(f(t)\) satisfies the identity \[f(t) - kf(1-t)=t\] for all values of \(t\). By replacing \(t\) with \(1-t\), determine \(f(t)\).
- Consider the new identity \[\begin{equation*} f(t) - f(1-t) = g(t). \hspace{1cm} \label{eq:star}\tag{$*$} \end{equation*}\]
Show that no function \(f(t)\) satisfies \(\eqref{eq:star}\) when \(g(t)=t\).
What condition must the function \(g(t)\) satisfy for there to be a solution \(f(t)\) to \(\eqref{eq:star}\)?
Find a solution \(f(t)\) to \(\eqref{eq:star}\) when \(g(t) = (2t-1)^3\).
