(i) Let $k \neq \pm 1$. The function $f ( t )$ satisfies the identity $$f ( t ) - k f ( 1 - t ) = t$$ for all values of $t$. By replacing $t$ with $1 - t$, determine $f ( t )$. (ii) Consider the new identity $$f ( t ) - f ( 1 - t ) = g ( t )$$ (a) Show that no function $f ( t )$ satisfies $( * )$ when $g ( t ) = t$. (b) What condition must the function $g ( t )$ satisfy for there to be a solution $f ( t )$ to $( * )$ ? (c) Find a solution $f ( t )$ to $( * )$ when $g ( t ) = ( 2 t - 1 ) ^ { 3 }$.
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\section*{2. For ALL APPLICANTS.}
(i) Let $k \neq \pm 1$. The function $f ( t )$ satisfies the identity
$$f ( t ) - k f ( 1 - t ) = t$$
for all values of $t$. By replacing $t$ with $1 - t$, determine $f ( t )$.\\
(ii) Consider the new identity
$$f ( t ) - f ( 1 - t ) = g ( t )$$
(a) Show that no function $f ( t )$ satisfies $( * )$ when $g ( t ) = t$.\\
(b) What condition must the function $g ( t )$ satisfy for there to be a solution $f ( t )$ to $( * )$ ?\\
(c) Find a solution $f ( t )$ to $( * )$ when $g ( t ) = ( 2 t - 1 ) ^ { 3 }$.