The function given is:

$f(x) = \sin(\pi x)$

Key Properties:

1. Domain: The function is defined for all real numbers, so its domain is:
   $(-\infty, \infty)$

2. Range: Since $\sin(\theta)$ always lies between $-1$ and $1$, we get:
   $-1 \leq f(x) \leq 1$ So, the range is $[-1,1]$.

3. Periodicity:

   • The sine function $\sin(\theta)$ has a fundamental period of $2\pi$.
   • Here, $\theta = \pi x$, so setting $\pi x = 2\pi$ gives the period:
     $x = 2$ This means $f(x)$ repeats every 2 units.

4. Intercepts:

   • x-intercepts: Solve $\sin(\pi x) = 0$, which happens when $\pi x = k\pi$, i.e., $x = k$, where $k$ is an integer.
   • y-intercept: At $x = 0$, we get $f(0) = \sin(0) = 0$, so the y-intercept is $(0,0)$.

5. Symmetry:

   • Since $\sin(\pi x)$ is odd (i.e., $f(-x) = -f(x)$), it has origin symmetry.

Would you like a graph or further explanations on any point?

Related Questions:

1. What is the derivative of $f(x) = \sin(\pi x)$ and its interpretation?
2. How do we find the maximum and minimum points of $f(x)$?
3. What happens if we modify the function to $f(x) = \sin(\pi x + \frac{\pi}{2})$?
4. How does the period change if we replace $\pi x$ with $2\pi x$?
5. What is the Fourier series representation of $f(x)$ over a given interval?

Tip:

To find where $f(x)$ is increasing or decreasing, take its derivative and analyze the sign!