Animated Solution for Mathematics - Functions: Which one is not periodic

Understanding Periodicity

Imagine standing on a beach, watching the waves roll in. Each wave is a perfect, repeating cycle. In mathematics, we call this phenomenon periodicity.

A function $f(x)$ is periodic if it repeats its values after a fixed interval $T$, such that $f(x+T)=f(x)$. This constant $T$ is the heartbeat of the function, its fundamental period.

When we look at complex functions, we are essentially looking for this heartbeat. If the heartbeat is constant, the function is periodic. If the rhythm changes, or if the waves never quite align, the function loses its periodicity.

When Waves Synchronize

In the JEE Advanced arena, you will often encounter functions that are sums of other functions, such as $f(x)=f_1(x)+f_2(x)$. To determine if the sum is periodic, we examine the individual periods $T_1$ and $T_2$.

The sum is periodic if and only if the ratio $\frac{T_1}{T_2}$ is a rational number. Think of it like two runners on a track. If one runner completes a lap in $2$ minutes and the other in $3$ minutes, they will meet again at the start line because the ratio $\frac{2}{3}$ is rational.

However, if one runner takes $1$ minute and the other takes $\pi$ minutes, they will never synchronize. This is the core logic behind checking expressions like $|\sin 3x|+{\sin }^{2}x$ or $\cos 4x+{\tan }^{2}x$. We calculate their periods, find the ratio, and if it is rational, we confirm the existence of a fundamental period.

The Case of $\cos \sqrt{x}$

Now, let us confront the villain of our problem: $\cos \sqrt{x}$. At first glance, it looks like a standard cosine wave, but the argument is $\sqrt{x}$.

As $x$ grows, $\sqrt{x}$ grows slower and slower. This means the frequency of the wave is decreasing, and the peaks are getting further apart. If you were to graph this, you would see the wave stretching out horizontally, never repeating the same pattern.

To prove this, we assume a period $T$ exists such that $\cos \sqrt{x+T}=\cos \sqrt{x}$. This leads us to the following condition:

By rationalizing the left side, we obtain:

As $x$ approaches infinity, the denominator grows without bound, forcing the left side to zero. Since $2n\pi$ is a constant, this creates a contradiction. Thus, $\cos \sqrt{x}$ is fundamentally non-periodic.

We have established that $\cos \sqrt{x}$ is non-periodic. When you add a periodic function like ${\cos }^{2}x$ to a non-periodic function like $\cos \sqrt{x}$, the non-periodicity dominates.

The sum cannot repeat because the non-periodic part will never return to its original value at the same time the periodic part does. Thus, the sum $\cos \sqrt{x}+{\cos }^{2}x$ is non-periodic.

By mastering this, you have learned to look past the surface of a function to see its true, underlying rhythm. Keep questioning and visualizing, and the mathematics will always reveal its secrets.