Question

If $f:\mathbb{R} \to \mathbb{R}$ is a function satisfying the property $f(x+1)+f(x+3)=2$ for all $x \in \mathbb{R}$, then $f$ is

• A. periodic with period 3
• B. periodic with period 4
• C. non-periodic
• D. periodic with period 5

Answer 1

Answer: (B)

periodic with period 4

Answer 2

The given functional equation is $f(x+1)+f(x+3)=2$ for all $x \in \mathbb{R}$.

To check for periodicity, we look for a positive real number $T$ such that $f(x+T) = f(x)$ for all $x \in \mathbb{R}$.

Replace $x$ with $x+1$: $f(x+2) + f(x+4) = 2$ for all $x \in \mathbb{R}$.

Replace $x$ with $x+2$: $f(x+3) + f(x+5) = 2$ for all $x \in \mathbb{R}$.

From equation (1), we can express $f(x+3)$ as: $f(x+3) = 2 - f(x+1)$

Substitute this expression for $f(x+3)$ into equation (3): $(2 - f(x+1)) + f(x+5) = 2$ $2 - f(x+1) + f(x+5) = 2$ $-f(x+1) + f(x+5) = 0$ $f(x+5) = f(x+1)$ for all $x \in \mathbb{R}$.

Let $y = x+1$. Then: $f(y+4) = f(y)$ for all $y \in \mathbb{R}$.

This equation $f(y+4) = f(y)$ shows that the function $f$ is periodic with period 4.