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  • How to use it?

  • Graphing y =:
  • -x^2-2x+1
  • 3x^4-4x^3
  • 3/2x^2-x^3
  • 3x^2-12x

  • Identical expressions

  • (ctg(pix)- one)^(one / three)
  • (ctg( Pi x) minus 1) to the power of (1 divide by 3)
  • (ctg( Pi x) minus one) to the power of (one divide by three)
  • (ctg(pix)-1)(1/3)
  • ctgpix-11/3
  • ctgpix-1^1/3
  • (ctg(pix)-1)^(1 divide by 3)

  • Similar expressions

  • (ctg(pix)+1)^(1/3)
Plotting a function graph/ (ctg(pix)-1)^(1/3)

Graphing y = (ctg(pix)-1)^(1/3)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       3 _______________
f(x) = \/ cot(pi*x) - 1
f(x)=cot⁡(πx)−13f{\left(x \right)} = \sqrt[3]{\cot{\left(\pi x \right)} - 1} 
f = (cot(pi*x) - 1)^(1/3)
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)−13=0\sqrt[3]{\cot{\left(\pi x \right)} - 1} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=14x_{1} = \frac{1}{4}
Numerical solution
x1=0.25x_{1} = 0.25
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)3(cot⁡(πx)−1)23=0\frac{\pi \left(- \cot^{2}{\left(\pi x \right)} - 1\right)}{3 \left(\cot{\left(\pi x \right)} - 1\right)^{\frac{2}{3}}} = 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(3cot⁡(πx)−cot⁡2(πx)+1cot⁡(πx)−1)(cot⁡2(πx)+1)9(cot⁡(πx)−1)23=0\frac{2 \pi^{2} \left(3 \cot{\left(\pi x \right)} - \frac{\cot^{2}{\left(\pi x \right)} + 1}{\cot{\left(\pi x \right)} - 1}\right) \left(\cot^{2}{\left(\pi x \right)} + 1\right)}{9 \left(\cot{\left(\pi x \right)} - 1\right)^{\frac{2}{3}}} = 0
Solve this equation
The roots of this equation
x1=acot⁡(34−174)πx_{1} = \frac{\operatorname{acot}{\left(\frac{3}{4} - \frac{\sqrt{17}}{4} \right)}}{\pi}
x2=acot⁡(34+174)πx_{2} = \frac{\operatorname{acot}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} \right)}}{\pi}

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
(−∞,acot⁡(34+174)π]\left(-\infty, \frac{\operatorname{acot}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} \right)}}{\pi}\right]
Convex at the intervals
[acot⁡(34+174)π,∞)\left[\frac{\operatorname{acot}{\left(\frac{3}{4} + \frac{\sqrt{17}}{4} \right)}}{\pi}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞cot⁡(πx)−13y = \lim_{x \to -\infty} \sqrt[3]{\cot{\left(\pi x \right)} - 1}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞cot⁡(πx)−13y = \lim_{x \to \infty} \sqrt[3]{\cot{\left(\pi x \right)} - 1}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (cot(pi*x) - 1)^(1/3), 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)−13x)y = x \lim_{x \to -\infty}\left(\frac{\sqrt[3]{\cot{\left(\pi x \right)} - 1}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(πx)−13x)y = x \lim_{x \to \infty}\left(\frac{\sqrt[3]{\cot{\left(\pi x \right)} - 1}}{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)−13=−cot⁡(πx)−13\sqrt[3]{\cot{\left(\pi x \right)} - 1} = \sqrt[3]{- \cot{\left(\pi x \right)} - 1}
- No
cot⁡(πx)−13=−−cot⁡(πx)−13\sqrt[3]{\cot{\left(\pi x \right)} - 1} = - \sqrt[3]{- \cot{\left(\pi x \right)} - 1}
- No
so, the function
not is
neither even, nor odd
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