Question

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

• A. $3$
• B. $5$
• C. $7$
• D. $11$

Answer 1

Answer: (B)

$5$

$f\left(x\right)=px+q$
$f\left(f\right(x\left)\right)=p(px+q)+q$
$=p^{2}x+q(p+1)$
$f\left(f\right(f\left(x\right)\left)\right)=p(p^{2}x+q(p+1\left)\right)+q$
$=p^{3}x+q\left(p\right(p+1)+1)$
$=8x+21$
By comparing coefficients, we get
$p^3=8$
$p=2$
And $q\left(2\right(3)+1)=21$
$7q=21$
$q=3$
Hence
$f\left(x\right)=2x+3$
Therefore, $p+q=2+3=5$