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Plotting a function graph/ ctg(pi*x)

Graphing y = ctg(pi*x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cot(pi*x)
f(x)=cot⁡(πx)f{\left(x \right)} = \cot{\left(\pi x \right)} 
f = cot(pi*x)
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
cot⁡(πx)=0\cot{\left(\pi x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=12x_{1} = \frac{1}{2}
Numerical solution
x1=0.5x_{1} = 0.5
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot(pi*x).
cot⁡(0π)\cot{\left(0 \pi \right)}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
π(−cot⁡2(πx)−1)=0\pi \left(- \cot^{2}{\left(\pi x \right)} - 1\right) = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2π2(cot⁡2(πx)+1)cot⁡(πx)=02 \pi^{2} \left(\cot^{2}{\left(\pi x \right)} + 1\right) \cot{\left(\pi x \right)} = 0
Solve this equation
The roots of this equation
x1=12x_{1} = \frac{1}{2}

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,12]\left(-\infty, \frac{1}{2}\right]
Convex at the intervals
[12,∞)\left[\frac{1}{2}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞cot⁡(πx)=−cot⁡(∞)\lim_{x \to -\infty} \cot{\left(\pi x \right)} = - \cot{\left(\infty \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−cot⁡(∞)y = - \cot{\left(\infty \right)}
lim⁡x→∞cot⁡(πx)=cot⁡(∞)\lim_{x \to \infty} \cot{\left(\pi x \right)} = \cot{\left(\infty \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=cot⁡(∞)y = \cot{\left(\infty \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(pi*x), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(cot⁡(πx)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\pi x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(πx)x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\pi x \right)}}{x}\right)
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
cot⁡(πx)=−cot⁡(πx)\cot{\left(\pi x \right)} = - \cot{\left(\pi x \right)}
- No
cot⁡(πx)=cot⁡(πx)\cot{\left(\pi x \right)} = \cot{\left(\pi x \right)}
- Yes
so, the function
is
odd
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