Vtyjkyuk

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 $t$. ($xin mathbb R$)

What I did:

$$f(2x+3)+f(2x+7)=2.........(1)$$

Replacing $x$ with $x-1$ in $(1)$,

$$f(2x+1)+f(2x+5)=2.........(2)$$

And replacing $x$ with $x+1$ in $(1)$,

$$f(2x+5)+f(2x+9)=2.........(3)$$

Subtracting $(2)$ from $(3)$, 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?

I found this tricky and I may have got it wrong, but I think that you are correct and the book is not.

If $f(2x+3)+f(2x+7)=2$ for all real $x$, then we can take $x=(y-3)/2$ to give
$$f(y)+f(y+4)=2tag$*$$$
and then, essentially following your argument,
$$f(y)=f(y+8)$$
for all real $y$. Now suppose that $f$ has period $t$: recall that this means $t>0$ and $f(y)=f(y+t)$ for all real $y$, and that no smaller positive $t$ has the same property. Suppose that there is an integer $k$ such that
$$kt<8<(k+1)t ;$$
this can be rewritten as
$$8=kt+t'quadhboxwithquad 0<t'<t .$$
Then we have
$$f(y)=f(y+8)=f(y+t'+kt)=f(y+t')quadhboxfor all $y$,$$
which contradicts the fact that $f$ has period $t$. Therefore we must have $8=kt$ for some positive integer $k$, that is, $t=8/k$.

Now $k$ cannot be even, for if $k=2m$ then for all $y$ we have $f(y)=f(y+mt)=f(y+4)$; in conjunction with $(*)$ this shows that $f(y)=1$ for all $y$, which is not (strictly speaking) periodic.

However it is possible that the period could be $t=8/k$ with $k$ odd. For example, take
$$f(y)=1+sinBigl(frac2pi ytBigr) .$$
Then, as is well known, $f$ has period $t$; also
$$f(y)+f(y+4)
=1+sinBigl(frac2pi ytBigr)+1+sinBigl(frac2pi yt+kpiBigr)=2$$
for all $y$, as required.

"How can I be sure that the t so found is the least?"

Answer : by finding an example where $8$ is the smallest period. For example, take $f$ such that $f(x)=fracx-8k2$ when $xin[8k,8k+4[$ and $f(x)=frac8k+8-x2$
when $xin[8k+4,8k+8[$, for every $kinmathbb Z$.

This creates a "sawtooth" function and only multiples of $8$ are periods.

Period is 4. It is correct. You got $f(2x+1)=f(2x+9)$, that means the period of $f(2x)$ is $8$. So period of $f(x)$ is $4$.

Since we have $f(2x) = f(2x + 8)$ for all $x$,
If we assume $g(x) = f(2x)$ then we have $g(x + 4) = g(x)$, and hence $g(x)$ is periodic with period $4$, not $f(x)$.

The period is $4$. Actually $f(x)$ and $f(2x)$ are different functions. You see, $cos(2x)$ and $cos(x)$ are different. The period of $cos(x)$ is $2pi$ but the period of $cos(2x)$ is $pi$. Similarly, the period of $f(x)$ is the the period of $f(2x)/2$.