Fermat 1997 Q23

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Problem Statement

if $f(x) = px + q$ and $f(f(f(x))) = 8x + 21$, and if $p, q \in \mathbb{R}$, then find $p+q$

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Solution

$$f(x) = px + q \; \; \; p, q \in \mathbb{R}$$ $$f(f(f(x))) = 8x + 21$$ $$f(f(px + q)) = 8x + 21$$ $$f(p(px + q) + q) = 8x + 21$$ $$p(p(px + q) + q) + q = 8x + 21$$ $$p^3x + p^2q + pq + q = 8x + 21$$ $$\implies p = 2$$ $$p^2q + pq + q = 21$$ $$4q + 2q + q = 21 \implies q = 3$$ $$\therefore p + q = 2+3 = 5$$