Let $\mathrm{f}(\mathrm{x})=3+2 \mathrm{x}$ and $\mathrm{gn}(\mathrm{x})=($ fo fo fo..... n times…
Let $\mathrm{f}(\mathrm{x})=3+2 \mathrm{x}$ and $\mathrm{gn}(\mathrm{x})=($ fo fo fo..... n times $)(\mathrm{x}) . \forall n \in \mathrm{~N}$ if all the lines $y=g_{\mathrm{n}}(x)$ pass through a fixed point $(\alpha, \beta)$, then $\alpha+\beta=$
$-5$
$-4$
$-3$
$-6$
Solution
$f(x)=3+2 x \Rightarrow g_1(x)=3+2 x$
$g_2(x)=f \circ f(x)=9+4 x$
$\mathrm{g}_3(x)=f \circ f \circ f(x)=21+8 x$
$g_n(x)=($ fofof $\ldots . . n$ times $)(x)=3\left(2^n-1\right)+2^n x$
For fixed point $x=y$
$\Rightarrow x=3\left(2^{\mathrm{n}}-1\right)+2^{\mathrm{n}} x \Rightarrow x=-3$
and $y=3\left(2^{\mathrm{n}}-1\right)+2^{\mathrm{n}} x \Rightarrow y=-3$
So, $(-3,-3)$ is a fixed point $\forall n \in N$
$\Rightarrow \alpha+\beta=-3-3=-6$