Review question

# When does $f$ satisfy the identity $f(t)-f(1-t)=g(t)$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9798

## Question

1. 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 $f(t)$.

2. Consider the new identity $\begin{equation*} f(t) - f(1-t) = g(t). \hspace{1cm} \label{eq:star}\tag{*} \end{equation*}$
1. Show that no function $f(t)$ satisfies $\eqref{eq:star}$ when $g(t)=t$.

2. What condition must the function $g(t)$ satisfy for there to be a solution $f(t)$ to $\eqref{eq:star}$?

3. Find a solution $f(t)$ to $\eqref{eq:star}$ when $g(t) = (2t-1)^3$.