Question

If $f\left(x\right)=px+q,f\left(f\right(f\left(x\right)\left)\right)=8x+21,$ where $p,q$ are real numbers, then $p+q=$

• A. $7$
• B. $12$
• C. $5$
• D. $13$

Answer 1

The correct option is C $5$

$f\left(x\right)=px+q\Rightarrow f\left(f\right(x\left)\right)=f(px+q)\Rightarrow f\left(f\right(x\left)\right)=p(px+q)+q\Rightarrow f\left(f\right(x\left)\right)=p^{2}x+pq+q\Rightarrow f\left(f\right(f\left(x\right)\left)\right)=p(p^{2}x+pq+q)+q8x+21=p^{3}x+p^{2}q+pq+q{p}^3=8\Rightarrow p=221=4q+2q+q\Rightarrow 7q=21\Rightarrow q=3\therefore p+q=5$