Q. If the function $f(x)$ is defined by $f(x) = a + bx$ and $f^{r} = fff\ldots$(repeated $r$ times), then $f^{r}(x)$ is equal to
A) $a + b^{r}x$
B) $ar + b^{r}x$
C) $ar + bx^{r}$
D) $a\left( \frac{b^{r} - 1}{b - 1} \right) + b^{r}x$
Explanation
(d)
$\because f(x) = a + bx$
$f\left\{ f(x) \right\} = a + b(a + bx)$
$= ab + a + b^{2}x = a(1 + b) + b^{2}x$
$f\lbrack f\left\{ f(x) \right\}\rbrack = f\{ a(1 + b) + b^{2}x\}$
$= a + b\{ a(1 + b) + b^{2}x\}$
$= a\left( 1 + b + b^{2} \right) + b^{3}x$
$\therefore f^{'}(x) = a\left( 1 + b + b^{2} + ... + b^{r - 1} \right) + b^{r}x$
$= a\left( \frac{b^{r} - 1}{b - 1} \right) + b^{r}x$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.