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Plotting a function graph/ ctg(x)+1

Graphing y = ctg(x)+1

v

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
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.0-5000050000
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=21.2057504117311x_{1} = 21.2057504117311
x2=−35.3429173528852x_{2} = -35.3429173528852
x3=−51.0508806208341x_{3} = -51.0508806208341
x4=2.35619449019234x_{4} = 2.35619449019234
x5=43.1968989868597x_{5} = 43.1968989868597
x6=−98.174770424681x_{6} = -98.174770424681
x7=−38.484510006475x_{7} = -38.484510006475
x8=11.7809724509617x_{8} = 11.7809724509617
x9=−79.3252145031423x_{9} = -79.3252145031423
x10=90.3207887907066x_{10} = 90.3207887907066
x11=30.6305283725005x_{11} = 30.6305283725005
x12=80.8960108299372x_{12} = 80.8960108299372
x13=46.3384916404494x_{13} = 46.3384916404494
x14=−88.7499924639117x_{14} = -88.7499924639117
x15=−7.06858347057703x_{15} = -7.06858347057703
x16=62.0464549083984x_{16} = 62.0464549083984
x17=36.9137136796801x_{17} = 36.9137136796801
x18=40.0553063332699x_{18} = 40.0553063332699
x19=52.621676947629x_{19} = 52.621676947629
x20=−66.7588438887831x_{20} = -66.7588438887831
x21=−63.6172512351933x_{21} = -63.6172512351933
x22=49.4800842940392x_{22} = 49.4800842940392
x23=65.1880475619882x_{23} = 65.1880475619882
x24=−16.4933614313464x_{24} = -16.4933614313464
x25=−95.0331777710912x_{25} = -95.0331777710912
x26=−10.2101761241668x_{26} = -10.2101761241668
x27=−32.2013246992954x_{27} = -32.2013246992954
x28=58.9048622548086x_{28} = 58.9048622548086
x29=−25.9181393921158x_{29} = -25.9181393921158
x30=−47.9092879672443x_{30} = -47.9092879672443
x31=−0.785398163397448x_{31} = -0.785398163397448
x32=96.6039740978861x_{32} = 96.6039740978861
x33=33.7721210260903x_{33} = 33.7721210260903
x34=27.4889357189107x_{34} = 27.4889357189107
x35=55.7632696012188x_{35} = 55.7632696012188
x36=87.1791961371168x_{36} = 87.1791961371168
x37=−73.0420291959627x_{37} = -73.0420291959627
x38=−41.6261026600648x_{38} = -41.6261026600648
x39=99.7455667514759x_{39} = 99.7455667514759
x40=−91.8915851175014x_{40} = -91.8915851175014
x41=84.037603483527x_{41} = 84.037603483527
x42=71.4712328691678x_{42} = 71.4712328691678
x43=77.7544181763474x_{43} = 77.7544181763474
x44=14.9225651045515x_{44} = 14.9225651045515
x45=24.3473430653209x_{45} = 24.3473430653209
x46=−82.4668071567321x_{46} = -82.4668071567321
x47=93.4623814442964x_{47} = 93.4623814442964
x48=−76.1836218495525x_{48} = -76.1836218495525
x49=68.329640215578x_{49} = 68.329640215578
x50=−60.4756585816035x_{50} = -60.4756585816035
x51=74.6128255227576x_{51} = 74.6128255227576
x52=−19.6349540849362x_{52} = -19.6349540849362
x53=−13.3517687777566x_{53} = -13.3517687777566
x54=−29.0597320457056x_{54} = -29.0597320457056
x55=5.49778714378214x_{55} = 5.49778714378214
x56=−69.9004365423729x_{56} = -69.9004365423729
x57=−22.776546738526x_{57} = -22.776546738526
x58=−57.3340659280137x_{58} = -57.3340659280137
x59=−44.7676953136546x_{59} = -44.7676953136546
x60=18.0641577581413x_{60} = 18.0641577581413
x61=8.63937979737193x_{61} = 8.63937979737193
x62=−3.92699081698724x_{62} = -3.92699081698724
x63=−54.1924732744239x_{63} = -54.1924732744239
x64=−85.6083998103219x_{64} = -85.6083998103219
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=1−cot⁡(x)\cot{\left(x \right)} + 1 = 1 - \cot{\left(x \right)}
- 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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