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Plotting a function graph/ y=ctgx-1

Graphing y = y=ctgx-1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cot(x) - 1
f(x)=cot⁡(x)−1f{\left(x \right)} = \cot{\left(x \right)} - 1 
f = cot(x) - 1
The graph of the function
02468-8-6-4-2-1010-10001000
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⁡(x)−1=0\cot{\left(x \right)} - 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π4x_{1} = \frac{\pi}{4}
Numerical solution
x1=−36.9137136796801x_{1} = -36.9137136796801
x2=51.0508806208341x_{2} = 51.0508806208341
x3=−74.6128255227576x_{3} = -74.6128255227576
x4=−84.037603483527x_{4} = -84.037603483527
x5=22.776546738526x_{5} = 22.776546738526
x6=3.92699081698724x_{6} = 3.92699081698724
x7=−96.6039740978861x_{7} = -96.6039740978861
x8=−43.1968989868597x_{8} = -43.1968989868597
x9=−33.7721210260903x_{9} = -33.7721210260903
x10=10.2101761241668x_{10} = 10.2101761241668
x11=16.4933614313464x_{11} = 16.4933614313464
x12=−8.63937979737193x_{12} = -8.63937979737193
x13=19.6349540849362x_{13} = 19.6349540849362
x14=44.7676953136546x_{14} = 44.7676953136546
x15=13.3517687777566x_{15} = 13.3517687777566
x16=47.9092879672443x_{16} = 47.9092879672443
x17=29.0597320457056x_{17} = 29.0597320457056
x18=60.4756585816035x_{18} = 60.4756585816035
x19=−99.7455667514759x_{19} = -99.7455667514759
x20=25.9181393921158x_{20} = 25.9181393921158
x21=85.6083998103219x_{21} = 85.6083998103219
x22=−80.8960108299372x_{22} = -80.8960108299372
x23=−11.7809724509617x_{23} = -11.7809724509617
x24=−62.0464549083984x_{24} = -62.0464549083984
x25=76.1836218495525x_{25} = 76.1836218495525
x26=69.9004365423729x_{26} = 69.9004365423729
x27=−2.35619449019234x_{27} = -2.35619449019234
x28=−68.329640215578x_{28} = -68.329640215578
x29=−55.7632696012188x_{29} = -55.7632696012188
x30=−65.1880475619882x_{30} = -65.1880475619882
x31=−14.9225651045515x_{31} = -14.9225651045515
x32=82.4668071567321x_{32} = 82.4668071567321
x33=−18.0641577581413x_{33} = -18.0641577581413
x34=63.6172512351933x_{34} = 63.6172512351933
x35=32.2013246992954x_{35} = 32.2013246992954
x36=54.1924732744239x_{36} = 54.1924732744239
x37=−40.0553063332699x_{37} = -40.0553063332699
x38=−49.4800842940392x_{38} = -49.4800842940392
x39=7.06858347057703x_{39} = 7.06858347057703
x40=98.174770424681x_{40} = 98.174770424681
x41=41.6261026600648x_{41} = 41.6261026600648
x42=−58.9048622548086x_{42} = -58.9048622548086
x43=0.785398163397448x_{43} = 0.785398163397448
x44=88.7499924639117x_{44} = 88.7499924639117
x45=−24.3473430653209x_{45} = -24.3473430653209
x46=−46.3384916404494x_{46} = -46.3384916404494
x47=−77.7544181763474x_{47} = -77.7544181763474
x48=−93.4623814442964x_{48} = -93.4623814442964
x49=57.3340659280137x_{49} = 57.3340659280137
x50=−87.1791961371168x_{50} = -87.1791961371168
x51=95.0331777710912x_{51} = 95.0331777710912
x52=−52.621676947629x_{52} = -52.621676947629
x53=−5.49778714378214x_{53} = -5.49778714378214
x54=79.3252145031423x_{54} = 79.3252145031423
x55=35.3429173528852x_{55} = 35.3429173528852
x56=−90.3207887907066x_{56} = -90.3207887907066
x57=38.484510006475x_{57} = 38.484510006475
x58=91.8915851175014x_{58} = 91.8915851175014
x59=−27.4889357189107x_{59} = -27.4889357189107
x60=73.0420291959627x_{60} = 73.0420291959627
x61=−21.2057504117311x_{61} = -21.2057504117311
x62=66.7588438887831x_{62} = 66.7588438887831
x63=−71.4712328691678x_{63} = -71.4712328691678
x64=−30.6305283725005x_{64} = -30.6305283725005
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
−cot⁡2(x)−1=0- \cot^{2}{\left(x \right)} - 1 = 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
2(cot⁡2(x)+1)cot⁡(x)=02 \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{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
(−∞,π2]\left(-\infty, \frac{\pi}{2}\right]
Convex at the intervals
[π2,∞)\left[\frac{\pi}{2}, \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⁡(x)−1)y = \lim_{x \to -\infty}\left(\cot{\left(x \right)} - 1\right)
True

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

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