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# If $f:R\rightarrow R$ is a function defined as $f(2x+3)+f(2x+7)=2$ $\forall x\in R$ then the period of $f(x)$ is

$\begin{array}{1 1} 2 \\ 4 \\ 8 \\ 12 \end{array}$

Toolbox:
• $f(x)$ is said to be a periodic function with period a if $f(x)=f(x+a)$
Ans (C)
Given $f(2x+3)+f(2x+7)=2$ $\forall x\in R$
$\Rightarrow f(y)+f(y+4)=2$..........(i) where $2x+3=y$
By replacing y by y-4 we get
$f(y-4)+f(y)=2$....................(ii)
On subtracting (i)-(ii) we get
$f(y+4)-f(y-4)=0$ $\forall y\in R$
Replacing $y \:\:by\:\: y+4$ we get
$f(y+8)=f(y)$
$\Rightarrow f$ is periodic function with period 8.
edited May 17, 2014