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1/(tan(x)-1)
Plotting a function graph/ 1/(tan(x)-1)

Graphing y = 1/(tan(x)-1)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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             1     
f(x) = 1*----------
         tan(x) - 1
f(x)=1⋅1tan⁡(x)−1f{\left(x \right)} = 1 \cdot \frac{1}{\tan{\left(x \right)} - 1} 
f = 1/(tan(x) - 1*1)
The graph of the function
0-90-80-70-60-50-40-30-20-1010-100100
The domain of the function
The points at which the function is not precisely defined:
x1=0.785398163397448x_{1} = 0.785398163397448
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⋅1tan⁡(x)−1=01 \cdot \frac{1}{\tan{\left(x \right)} - 1} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=−95.8185759344887x_{1} = -95.8185759344887
x2=−73.8274273593601x_{2} = -73.8274273593601
x3=−86.3937979737193x_{3} = -86.3937979737193
x4=29.845130209103x_{4} = 29.845130209103
x5=−80.1106126665397x_{5} = -80.1106126665397
x6=98.9601685880785x_{6} = 98.9601685880785
x7=36.1283155162826x_{7} = 36.1283155162826
x8=95.8185759344887x_{8} = 95.8185759344887
x9=32.9867228626928x_{9} = 32.9867228626928
x10=−32.9867228626928x_{10} = -32.9867228626928
x11=−39.2699081698724x_{11} = -39.2699081698724
x12=−23.5619449019235x_{12} = -23.5619449019235
x13=−29.845130209103x_{13} = -29.845130209103
x14=−48.6946861306418x_{14} = -48.6946861306418
x15=−36.1283155162826x_{15} = -36.1283155162826
x16=45.553093477052x_{16} = 45.553093477052
x17=−83.2522053201295x_{17} = -83.2522053201295
x18=−20.4203522483337x_{18} = -20.4203522483337
x19=17.2787595947439x_{19} = 17.2787595947439
x20=−10.9955742875643x_{20} = -10.9955742875643
x21=70.6858347057703x_{21} = 70.6858347057703
x22=67.5442420521806x_{22} = 67.5442420521806
x23=−7.85398163397448x_{23} = -7.85398163397448
x24=−45.553093477052x_{24} = -45.553093477052
x25=−76.9690200129499x_{25} = -76.9690200129499
x26=−61.261056745001x_{26} = -61.261056745001
x27=−54.9778714378214x_{27} = -54.9778714378214
x28=−1.5707963267949x_{28} = -1.5707963267949
x29=1.5707963267949x_{29} = 1.5707963267949
x30=89.5353906273091x_{30} = 89.5353906273091
x31=−14.1371669411541x_{31} = -14.1371669411541
x32=−89.5353906273091x_{32} = -89.5353906273091
x33=51.8362787842316x_{33} = 51.8362787842316
x34=−17.2787595947439x_{34} = -17.2787595947439
x35=−42.4115008234622x_{35} = -42.4115008234622
x36=−92.6769832808989x_{36} = -92.6769832808989
x37=14.1371669411541x_{37} = 14.1371669411541
x38=73.8274273593601x_{38} = 73.8274273593601
x39=−67.5442420521806x_{39} = -67.5442420521806
x40=−98.9601685880785x_{40} = -98.9601685880785
x41=61.261056745001x_{41} = 61.261056745001
x42=42.4115008234622x_{42} = 42.4115008234622
x43=20.4203522483337x_{43} = 20.4203522483337
x44=83.2522053201295x_{44} = 83.2522053201295
x45=26.7035375555132x_{45} = 26.7035375555132
x46=76.9690200129499x_{46} = 76.9690200129499
x47=58.1194640914112x_{47} = 58.1194640914112
x48=10.9955742875643x_{48} = 10.9955742875643
x49=54.9778714378214x_{49} = 54.9778714378214
x50=92.6769832808989x_{50} = 92.6769832808989
x51=86.3937979737193x_{51} = 86.3937979737193
x52=−102.101761241668x_{52} = -102.101761241668
x53=64.4026493985908x_{53} = 64.4026493985908
x54=−64.4026493985908x_{54} = -64.4026493985908
x55=−26.7035375555132x_{55} = -26.7035375555132
x56=4.71238898038469x_{56} = 4.71238898038469
x57=−70.6858347057703x_{57} = -70.6858347057703
x58=−58.1194640914112x_{58} = -58.1194640914112
x59=23.5619449019235x_{59} = 23.5619449019235
x60=7.85398163397448x_{60} = 7.85398163397448
x61=39.2699081698724x_{61} = 39.2699081698724
x62=48.6946861306418x_{62} = 48.6946861306418
x63=−4.71238898038469x_{63} = -4.71238898038469
x64=−51.8362787842316x_{64} = -51.8362787842316
x65=80.1106126665397x_{65} = 80.1106126665397
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 1/(tan(x) - 1*1).
1⋅1(−1)1+tan⁡(0)1 \cdot \frac{1}{\left(-1\right) 1 + \tan{\left(0 \right)}}
The result:
f(0)=−1f{\left(0 \right)} = -1
The point:
(0, -1)
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
−tan⁡2(x)−1(tan⁡(x)−1)2=0\frac{- \tan^{2}{\left(x \right)} - 1}{\left(\tan{\left(x \right)} - 1\right)^{2}} = 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(tan⁡(x)−tan⁡2(x)+1tan⁡(x)−1)(tan⁡2(x)+1)(tan⁡(x)−1)2=0- \frac{2 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} - 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} - 1\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0.785398163397448x_{1} = 0.785398163397448

lim⁡x→0.785398163397448−(−2(tan⁡(x)−tan⁡2(x)+1tan⁡(x)−1)(tan⁡2(x)+1)(tan⁡(x)−1)2)=−5.84600654932361⋅1048\lim_{x \to 0.785398163397448^-}\left(- \frac{2 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} - 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} - 1\right)^{2}}\right) = -5.84600654932361 \cdot 10^{48}
Let's take the limit
lim⁡x→0.785398163397448+(−2(tan⁡(x)−tan⁡2(x)+1tan⁡(x)−1)(tan⁡2(x)+1)(tan⁡(x)−1)2)=−5.84600654932361⋅1048\lim_{x \to 0.785398163397448^+}\left(- \frac{2 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} - 1}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} - 1\right)^{2}}\right) = -5.84600654932361 \cdot 10^{48}
Let's take the limit
- limits are equal, then skip the corresponding point

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
(−∞,−π4]\left(-\infty, - \frac{\pi}{4}\right]
Convex at the intervals
[−π4,∞)\left[- \frac{\pi}{4}, \infty\right)
Vertical asymptotes
Have:
x1=0.785398163397448x_{1} = 0.785398163397448
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(1⋅1tan⁡(x)−1)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(1 \cdot \frac{1}{\tan{\left(x \right)} - 1}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(1⋅1tan⁡(x)−1)=⟨−∞,∞⟩\lim_{x \to \infty}\left(1 \cdot \frac{1}{\tan{\left(x \right)} - 1}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 1/(tan(x) - 1*1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(1x(tan⁡(x)−1))=lim⁡x→−∞(1x(tan⁡(x)−1))\lim_{x \to -\infty}\left(\frac{1}{x \left(\tan{\left(x \right)} - 1\right)}\right) = \lim_{x \to -\infty}\left(\frac{1}{x \left(\tan{\left(x \right)} - 1\right)}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(1x(tan⁡(x)−1))y = x \lim_{x \to -\infty}\left(\frac{1}{x \left(\tan{\left(x \right)} - 1\right)}\right)
lim⁡x→∞(1x(tan⁡(x)−1))=lim⁡x→∞(1x(tan⁡(x)−1))\lim_{x \to \infty}\left(\frac{1}{x \left(\tan{\left(x \right)} - 1\right)}\right) = \lim_{x \to \infty}\left(\frac{1}{x \left(\tan{\left(x \right)} - 1\right)}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(1x(tan⁡(x)−1))y = x \lim_{x \to \infty}\left(\frac{1}{x \left(\tan{\left(x \right)} - 1\right)}\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⋅1tan⁡(x)−1=1−tan⁡(x)−11 \cdot \frac{1}{\tan{\left(x \right)} - 1} = \frac{1}{- \tan{\left(x \right)} - 1}
- No
1⋅1tan⁡(x)−1=−1−tan⁡(x)−11 \cdot \frac{1}{\tan{\left(x \right)} - 1} = - \frac{1}{- \tan{\left(x \right)} - 1}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = 1/(tan(x)-1)
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