I have a problem in finding the period of functions given in the form of functional equations.
Q. If $f(x)$ is periodic with period $t$ such that $f(2x+3)+f(2x+7)=2$. Find $t$. ($x\in \mathbb R$)
What I did:
Replacing $x$ with $x-1$ in $(1)$,
And replacing $x$ with $x+1$ in $(1)$,
Subtracting $(2)$ and $(3)$, I get $$f(2x+1)=f(2x+9)$$
Since $x \in \mathbb R \iff 2x \in \mathbb R$, replace $2x$ with $x$ to get$$f(x)=f(x+8)\implies t=8$$
But sadly, my textbook's answer is $t=4$.
Is my method correct? How can I be sure that the $t$ so found is the least?