This question stemmed from a larger question:
Determine if the following expression is periodic or aperiodic. If periodic, state it's functional period.
$$f(x) = 2\sin(\frac{2\pi{x}}{7}) - \cos(\frac{3\pi{x}}{5}) + \tan(\frac{3\pi{x}}{2}) - 3x + 3[x] + 4$$
We can further express it as
$$f(x) = 2\sin(\frac{2\pi{x}}{7}) - \cos(\frac{3\pi{x}}{5}) + \tan(\frac{3\pi{x}}{2}) - 3\{ x \} + 4$$
We know that,
Period of $y = 2\sin(\frac{2\pi{x}}{7})$ - $7$
Period of $y = \cos(\frac{3\pi{x}}{5})$ - $\frac{10}{3}$
Period of $y = \tan(\frac{3\pi{x}}{2})$ - $\frac{2}{3}$
Period of $y = 3\{ x \}$ - $1$
But what is the period of $y = 4$? Shouldn't it be indeterminate, since the function repeats itself after every iteration, and we can't determine the gap between any 2 iterations?
To determine the fundamental period of $f(x)$, we need to know the periods of all the functions in $f(x)$, and take their LCM.
It's quite counter intuitive, since we know $y = 4$ is periodic, yet we can't determine it's period.
I plugged $f(x)$ into Desmos, and it gave me this, leading me to believe $f(x)$ was aperiodic..
This is the function I plugged into Desmos: $2\sin\left(\frac{2\pi x}{7}\right)\ -\ \cos\left(\frac{3\pi x}{5}\right)\ +\ \tan\left(\frac{3\pi x}{2}\right)\ -\ 3\operatorname{mod}\left(x,1\right)\ +4$
However, in an attempt to salvage the situation, I decided to neglect the $y =4 $ term and calculate the fundamental period, which I was getting as $70$ (LCM of $7, \frac{10}{3}, \frac{2}{3}, 1)$.
I checked the graph again, and it seems to be correct!
Graph when $x = 0$: graph at x = 0
Graph when $x = 70$: graph at x = 70
Here's my question: What is the period of $y = 4$, or $y = k$, where k is any constant?
$y=k$ is periodic with any period that you wish. When you are combining periodic functions and looking for the period you can ignore it. Compare $\sin x$ and $1+ \sin x$. The period of both is $2\pi$.
In your case you need a period that is a multiple of all the individual periods. The smallest one is $70$ so your graph is not long enough to show a period. It is a hard function to plot because the tangents go off to infinity so often. I doubt that if you plot over a range of $150$ you will be able to see the periodicity by eye, but you can try.