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abs(1/ctg(x))
Plotting a function graph/ abs(1/ctg(x))

Graphing y = abs(1/ctg(x))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       |    1   |
f(x) = |1*------|
       |  cot(x)|
f(x)=∣1⋅1cot⁡(x)∣f{\left(x \right)} = \left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right| 
f = Abs(1/cot(x))
The graph of the function
0-50-40-30-20-101020304050600100
The domain of the function
The points at which the function is not precisely defined:
x1=1.5707963267949x_{1} = 1.5707963267949
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:
∣1⋅1cot⁡(x)∣=0\left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right| = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=69.1150383789755x_{1} = 69.1150383789755
x2=−50.2654824574367x_{2} = -50.2654824574367
x3=−25.1327412287183x_{3} = -25.1327412287183
x4=87.9645943005142x_{4} = 87.9645943005142
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=97.3893722612836x_{6} = 97.3893722612836
x7=−81.6814089933346x_{7} = -81.6814089933346
x8=−21.9911485751286x_{8} = -21.9911485751286
x9=62.8318530717959x_{9} = 62.8318530717959
x10=−69.1150383789755x_{10} = -69.1150383789755
x11=81.6814089933346x_{11} = 81.6814089933346
x12=−56.5486677646163x_{12} = -56.5486677646163
x13=37.6991118430775x_{13} = 37.6991118430775
x14=25.1327412287183x_{14} = 25.1327412287183
x15=−78.5398163397448x_{15} = -78.5398163397448
x16=−62.8318530717959x_{16} = -62.8318530717959
x17=−65.9734457253857x_{17} = -65.9734457253857
x18=−87.9645943005142x_{18} = -87.9645943005142
x19=−53.4070751110265x_{19} = -53.4070751110265
x20=21.9911485751286x_{20} = 21.9911485751286
x21=−47.1238898038469x_{21} = -47.1238898038469
x22=−100.530964914873x_{22} = -100.530964914873
x23=6.28318530717959x_{23} = 6.28318530717959
x24=−75.398223686155x_{24} = -75.398223686155
x25=−72.2566310325652x_{25} = -72.2566310325652
x26=9.42477796076938x_{26} = 9.42477796076938
x27=56.5486677646163x_{27} = 56.5486677646163
x28=65.9734457253857x_{28} = 65.9734457253857
x29=100.530964914873x_{29} = 100.530964914873
x30=−28.2743338823081x_{30} = -28.2743338823081
x31=43.9822971502571x_{31} = 43.9822971502571
x32=−6.28318530717959x_{32} = -6.28318530717959
x33=31.4159265358979x_{33} = 31.4159265358979
x34=−3.14159265358979x_{34} = -3.14159265358979
x35=28.2743338823081x_{35} = 28.2743338823081
x36=−37.6991118430775x_{36} = -37.6991118430775
x37=−84.8230016469244x_{37} = -84.8230016469244
x38=84.8230016469244x_{38} = 84.8230016469244
x39=−34.5575191894877x_{39} = -34.5575191894877
x40=−91.106186954104x_{40} = -91.106186954104
x41=15.707963267949x_{41} = 15.707963267949
x42=94.2477796076938x_{42} = 94.2477796076938
x43=75.398223686155x_{43} = 75.398223686155
x44=34.5575191894877x_{44} = 34.5575191894877
x45=91.106186954104x_{45} = 91.106186954104
x46=−18.8495559215388x_{46} = -18.8495559215388
x47=−12.5663706143592x_{47} = -12.5663706143592
x48=−15.707963267949x_{48} = -15.707963267949
x49=59.6902604182061x_{49} = 59.6902604182061
x50=−94.2477796076938x_{50} = -94.2477796076938
x51=47.1238898038469x_{51} = 47.1238898038469
x52=−43.9822971502571x_{52} = -43.9822971502571
x53=−9.42477796076938x_{53} = -9.42477796076938
x54=3.14159265358979x_{54} = 3.14159265358979
x55=40.8407044966673x_{55} = 40.8407044966673
x56=−40.8407044966673x_{56} = -40.8407044966673
x57=72.2566310325652x_{57} = 72.2566310325652
x58=53.4070751110265x_{58} = 53.4070751110265
x59=78.5398163397448x_{59} = 78.5398163397448
x60=50.2654824574367x_{60} = 50.2654824574367
x61=−97.3893722612836x_{61} = -97.3893722612836
x62=12.5663706143592x_{62} = 12.5663706143592
x63=−31.4159265358979x_{63} = -31.4159265358979
x64=18.8495559215388x_{64} = 18.8495559215388
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to Abs(1/cot(x)).
∣1⋅1cot⁡(0)∣\left|{1 \cdot \frac{1}{\cot{\left(0 \right)}}}\right|
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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)+1cot⁡2(x)sign⁡(cot⁡(x))=0\frac{\cot^{2}{\left(x \right)} + 1}{\cot^{2}{\left(x \right)} \operatorname{sign}{\left(\cot{\left(x \right)} \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
2(cot⁡2(x)+1)((cot⁡2(x)+1)δ(cot⁡(x))cot⁡(x)sign⁡(cot⁡(x))+cot⁡2(x)+1cot⁡2(x)−1)cot⁡(x)sign⁡(cot⁡(x))=0\frac{2 \left(\cot^{2}{\left(x \right)} + 1\right) \left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \delta\left(\cot{\left(x \right)}\right)}{\cot{\left(x \right)} \operatorname{sign}{\left(\cot{\left(x \right)} \right)}} + \frac{\cot^{2}{\left(x \right)} + 1}{\cot^{2}{\left(x \right)}} - 1\right)}{\cot{\left(x \right)} \operatorname{sign}{\left(\cot{\left(x \right)} \right)}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=1.5707963267949x_{1} = 1.5707963267949
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞∣1⋅1cot⁡(x)∣=lim⁡x→−∞∣1⋅1cot⁡(x)∣\lim_{x \to -\infty} \left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right| = \lim_{x \to -\infty} \left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right|
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞∣1⋅1cot⁡(x)∣y = \lim_{x \to -\infty} \left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right|
lim⁡x→∞∣1⋅1cot⁡(x)∣=lim⁡x→∞∣1⋅1cot⁡(x)∣\lim_{x \to \infty} \left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right| = \lim_{x \to \infty} \left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right|
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞∣1⋅1cot⁡(x)∣y = \lim_{x \to \infty} \left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right|
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(1/cot(x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(∣1⋅1cot⁡(x)∣x)=lim⁡x→−∞(∣1⋅1cot⁡(x)∣x)\lim_{x \to -\infty}\left(\frac{\left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right|}{x}\right) = \lim_{x \to -\infty}\left(\frac{\left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right|}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(∣1⋅1cot⁡(x)∣x)y = x \lim_{x \to -\infty}\left(\frac{\left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right|}{x}\right)
lim⁡x→∞(∣1⋅1cot⁡(x)∣x)=lim⁡x→∞(∣1⋅1cot⁡(x)∣x)\lim_{x \to \infty}\left(\frac{\left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right|}{x}\right) = \lim_{x \to \infty}\left(\frac{\left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right|}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(∣1⋅1cot⁡(x)∣x)y = x \lim_{x \to \infty}\left(\frac{\left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\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:
∣1⋅1cot⁡(x)∣=1∣cot⁡(x)∣\left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right| = \frac{1}{\left|{\cot{\left(x \right)}}\right|}
- No
∣1⋅1cot⁡(x)∣=−1∣cot⁡(x)∣\left|{1 \cdot \frac{1}{\cot{\left(x \right)}}}\right| = - \frac{1}{\left|{\cot{\left(x \right)}}\right|}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = abs(1/ctg(x))
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