Question

If a polynomial function $f\left(x\right)$ satisfies $f\left(f\left(f\left(x\right)\right)\right)=8x+21$, then find $f\left(1\right)$

Answer 1

$f\left(f\left(f\left(x\right)\right)\right)=8x+21$

$f\left(x\right)$ i s a linear polynomial function.

$f\left(x\right)=ax+b$

$f\left(f\left(x\right)\right)=a\left(f\left(x\right)\right)+b$

$=a(ax+b)+b$

$=a^{2}x+ab+b$

$f\left(f\left(f\left(x\right)\right)\right)=a(a^{2}x+ab+b)+b$

$8x+21=a^{3}x+a^{2}b+ab+b$

$a^2=8$

$a=2$

$21=a^{2}b+ab+b$

$21=4b+2b+b$

$21=6b+b$

$7b=21$

$b=3$

$f\left(x\right)=ax+b$

$=2x+b$

$f\left(1\right)=2\times 1+3$

$f\left(1\right)=5$