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(D) function $f$ is periodic with period $T$ if $f(x+T) = f(x)$ for all $x$. For the sum $f+g$ to be periodic, there must exist a constant $T > 0$ suc

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periodic if $\frac{T_{1}}{T_{2}}$ is a rational number

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(D) function $f$ is periodic with period $T$ if $f(x+T) = f(x)$ for all $x$. For the sum $f+g$ to be periodic, there must exist a constant $T > 0$ such that $(f+g)(x+T) = (f+g)(x)$. This implies $f(x+T) + g(x+T) = f(x) + g(x)$. If $\frac{T_{1}}{T_{2}}$ is a rational number, say $\frac{T_{1}}{T_{2}} = \frac{p}{q}$ where $p, q \in \mathbb{Z}^{+}$, then $qT_{1} = pT_{2} = T$. In this case, $f(x+T) = f(x)$ and $g(x+T) = g(x)$, so $(f+g)(x+T) = f(x) + g(x)$. Thus, $f+g$ is periodic if the ratio of their periods is a rational number.

Let $f$ and $g$ be periodic functions with the periods $T_{1}$ and $T_{2}$ respectively. Then $f+g$ is

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