Question

If f:rr is a function satisfying the property f (2x+3) + f (2x+7) = 2, xr, then the period of f(x) is

Answer 1

Q. If f(x) is periodic with period t such that f(2x+3)+f(2x+7)=2. Find t. (x∈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∈R⟺2x∈R, replace 2x with x to get

f(x)=f(x+8)⟹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?

Answer 2

Step-by-step explanation:

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

put x=x+2

f(2(x+2)+3)+f(2(x+2)+7)=2

f(2x+7)+f(2x+11)=2.......(2)

(2) - (1)

f(2x+3)-f(2x+11)=0

f(2x+2)=f(2x+11)

f(2x+3)=f(2(x+4)+3)

f(x)=f(x+4)

Therefore the period of f(x) is 4

I hope this may help you