Question

Let $g$ : $R\to R$ be given by $g(x)=3+4x$. If $g^n(x)=gogo$....$og(x)$ and $g^n(x)=A+Bx$, then $A$ and $B$ are

• A. $2^{n+1}-1$, $2^{n+1}$
• B. $4^n-1$, $4^n$
• C. $3^n$, $3^n+1$
• D. None of these

Answer 1

Answer: (B)

Since $g(x)=3+4x$
$\therefore g^2\left(x\right)=\left(gog\right)\left(x\right)=g\left\{g\left(x\right)\right\}$
$=g\left(3+4x\right)=3+4\left(3+4x\right)$
or $g^2\left(x\right)=15+4^{2}x=\left(4^2-1\right)+4^{2}x$
Now $g^3\left(x\right)=\left(gogog\right)x=g\left\{g^2\left(x\right)\right\}$
$=g\left(15+4^{2}x\right)=3+4\left(15+4^{2}x\right)$
$=63+4^{3}x=\left(4^3-1\right)+4^{3}x$
Similarly, we get $g^n\left(x\right)=\left(4^n-1\right)+4^{n}x$