The fundamental period of $f(x) = \{ x \} + \cos(\pi x)$ is determined by analyzing the periods of the individual components:

1. Fractional part function, ( { x }:
   The fractional part function, $\{ x \}$, is defined as: $\{ x \} = x - \lfloor x \rfloor,$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.
   $\{ x \}$ is periodic with period $T_1 = 1$.

2. Cosine function, ( \cos(\pi x):
   The cosine function, $\cos(\pi x)$, has a period $T_2 = \frac{2\pi}{\pi} = 2$.

Fundamental Period of $f(x)$:

The fundamental period of $f(x)$ is the least common multiple (LCM) of the individual periods $T_1 = 1$ and $T_2 = 2$.

• Since $\text{LCM}(1, 2) = 2$, the fundamental period of $f(x)$ is: $T = 2.$

Verification of Periodicity:

For $T = 2$, we verify if $f(x + 2) = f(x)$: $f(x + 2) = \{ x + 2 \} + \cos(\pi(x + 2)).$

• $\{ x + 2 \} = (x + 2) - \lfloor x + 2 \rfloor = \{ x \}$,
• $\cos(\pi(x + 2)) = \cos(\pi x + 2\pi) = \cos(\pi x)$ (since cosine is $2\pi$-periodic).

Thus: $f(x + 2) = \{ x \} + \cos(\pi x) = f(x).$

Therefore, the fundamental period of $f(x)$ is $\boxed{2}$.

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Related Questions:

1. What is the period of the function $g(x) = \sin(3x) + \cos(5x)$?
2. How does the period of the fractional part function $\{ x \}$ differ from $x \mod 1$?
3. Why is the LCM used to find the fundamental period of a sum of periodic functions?
4. How does $\lfloor x \rfloor$ affect the periodicity of functions involving $\{ x \}$?
5. Can you graph $f(x) = \{ x \} + \cos(\pi x)$ to visualize its periodicity?

Tip: To determine the fundamental period of a sum of periodic functions, always confirm that the individual components are periodic and check for their compatibility in terms of alignment at multiples of the periods.