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Q.
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
Relations and Functions - Part 2
Solution:
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 $g^{n}\left(x\right)=\left(4^{n}-1\right)+4^{n}x$