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Plotting a function graph/ tan(tan(2*x))

Graphing y = tan(tan(2*x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = tan(tan(2*x))
f(x)=tan⁡(tan⁡(2x))f{\left(x \right)} = \tan{\left(\tan{\left(2 x \right)} \right)} 
f = tan(tan(2*x))
The graph of the function
02468-8-6-4-2-1010-250250
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:
tan⁡(tan⁡(2x))=0\tan{\left(\tan{\left(2 x \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−97.3893722612836x_{1} = -97.3893722612836
x2=−43.9822971502571x_{2} = -43.9822971502571
x3=−29.845130209103x_{3} = -29.845130209103
x4=−84.0058153926878x_{4} = -84.0058153926878
x5=−40.0235182424307x_{5} = -40.0235182424307
x6=56.5486677646163x_{6} = 56.5486677646163
x7=26.0722239276738x_{7} = 26.0722239276738
x8=28.2743338823081x_{8} = 28.2743338823081
x9=29.845130209103x_{9} = 29.845130209103
x10=−81.6814089933346x_{10} = -81.6814089933346
x11=−15.707963267949x_{11} = -15.707963267949
x12=72.2566310325652x_{12} = 72.2566310325652
x13=95.8185759344887x_{13} = 95.8185759344887
x14=68.2507246204339x_{14} = 68.2507246204339
x15=76.9690200129499x_{15} = 76.9690200129499
x16=−53.4070751110265x_{16} = -53.4070751110265
x17=−85.687315405466x_{17} = -85.687315405466
x18=40.1081602314601x_{18} = 40.1081602314601
x19=100.530964914873x_{19} = 100.530964914873
x20=−9.42477796076938x_{20} = -9.42477796076938
x21=81.6814089933346x_{21} = 81.6814089933346
x22=70.6858347057703x_{22} = 70.6858347057703
x23=−21.9911485751286x_{23} = -21.9911485751286
x24=−25.7640548565578x_{24} = -25.7640548565578
x25=−14.1371669411541x_{25} = -14.1371669411541
x26=42.4115008234622x_{26} = 42.4115008234622
x27=−72.2566310325652x_{27} = -72.2566310325652
x28=10.2630300223571x_{28} = 10.2630300223571
x29=−55.7433857219175x_{29} = -55.7433857219175
x30=−47.7552034316864x_{30} = -47.7552034316864
x31=15.707963267949x_{31} = 15.707963267949
x32=50.2654824574367x_{32} = 50.2654824574367
x33=−28.2743338823081x_{33} = -28.2743338823081
x34=−6.28318530717959x_{34} = -6.28318530717959
x35=−105.243353895258x_{35} = -105.243353895258
x36=−17.2787595947439x_{36} = -17.2787595947439
x37=62.2005394439564x_{37} = 62.2005394439564
x38=−80.1106126665397x_{38} = -80.1106126665397
x39=−87.9645943005142x_{39} = -87.9645943005142
x40=0x_{40} = 0
x41=20.4203522483337x_{41} = 20.4203522483337
x42=18.2182422936993x_{42} = 18.2182422936993
x43=−65.9734457253857x_{43} = -65.9734457253857
x44=−73.8274273593601x_{44} = -73.8274273593601
x45=6.28318530717959x_{45} = 6.28318530717959
x46=−67.5442420521806x_{46} = -67.5442420521806
x47=−36.1283155162826x_{47} = -36.1283155162826
x48=94.2477796076938x_{48} = 94.2477796076938
x49=65.9734457253857x_{49} = 65.9734457253857
x50=−11.8598880461058x_{50} = -11.8598880461058
x51=36.1283155162826x_{51} = 36.1283155162826
x52=32.9867228626928x_{52} = 32.9867228626928
x53=21.9911485751286x_{53} = 21.9911485751286
x54=54.1135576792799x_{54} = 54.1135576792799
x55=78.5398163397448x_{55} = 78.5398163397448
x56=−89.5353906273091x_{56} = -89.5353906273091
x57=−62.0993088065887x_{57} = -62.0993088065887
x58=7.85398163397448x_{58} = 7.85398163397448
x59=14.1371669411541x_{59} = 14.1371669411541
x60=86.3937979737193x_{60} = 86.3937979737193
x61=3.95877890782638x_{61} = 3.95877890782638
x62=87.9645943005142x_{62} = 87.9645943005142
x63=12.5663706143592x_{63} = 12.5663706143592
x64=−3.14159265358979x_{64} = -3.14159265358979
x65=−63.6437522155381x_{65} = -63.6437522155381
x66=51.8362787842316x_{66} = 51.8362787842316
x67=−37.6991118430775x_{67} = -37.6991118430775
x68=46.1844071048915x_{68} = 46.1844071048915
x69=−51.8362787842316x_{69} = -51.8362787842316
x70=90.4748733262645x_{70} = 90.4748733262645
x71=−1.5707963267949x_{71} = -1.5707963267949
x72=59.6902604182061x_{72} = 59.6902604182061
x73=−23.5619449019235x_{73} = -23.5619449019235
x74=64.4026493985908x_{74} = 64.4026493985908
x75=43.9822971502571x_{75} = 43.9822971502571
x76=−50.2654824574367x_{76} = -50.2654824574367
x77=37.6991118430775x_{77} = 37.6991118430775
x78=−95.8185759344887x_{78} = -95.8185759344887
x79=80.1106126665397x_{79} = 80.1106126665397
x80=73.8274273593601x_{80} = 73.8274273593601
x81=−59.6902604182061x_{81} = -59.6902604182061
x82=−7.85398163397448x_{82} = -7.85398163397448
x83=−77.7809191566922x_{83} = -77.7809191566922
x84=−75.398223686155x_{84} = -75.398223686155
x85=−69.7463520068149x_{85} = -69.7463520068149
x86=23.5619449019235x_{86} = 23.5619449019235
x87=92.6769832808989x_{87} = 92.6769832808989
x88=−91.7375005819435x_{88} = -91.7375005819435
x89=−45.553093477052x_{89} = -45.553093477052
x90=−33.6180364905323x_{90} = -33.6180364905323
x91=83.2522053201295x_{91} = 83.2522053201295
x92=−19.5560384897921x_{92} = -19.5560384897921
x93=1.5707963267949x_{93} = 1.5707963267949
x94=−58.1194640914112x_{94} = -58.1194640914112
x95=58.1194640914112x_{95} = 58.1194640914112
x96=−94.2477796076938x_{96} = -94.2477796076938
x97=48.0633725028023x_{97} = 48.0633725028023
x98=98.9601685880785x_{98} = 98.9601685880785
x99=34.5575191894877x_{99} = 34.5575191894877
x100=−31.4159265358979x_{100} = -31.4159265358979
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(tan(2*x)).
tan⁡(tan⁡(0⋅2))\tan{\left(\tan{\left(0 \cdot 2 \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
(2tan⁡2(2x)+2)(tan⁡2(tan⁡(2x))+1)=0\left(2 \tan^{2}{\left(2 x \right)} + 2\right) \left(\tan^{2}{\left(\tan{\left(2 x \right)} \right)} + 1\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
8((tan⁡2(2x)+1)tan⁡(tan⁡(2x))+tan⁡(2x))(tan⁡2(2x)+1)(tan⁡2(tan⁡(2x))+1)=08 \left(\left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(\tan{\left(2 x \right)} \right)} + \tan{\left(2 x \right)}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(2 x \right)} \right)} + 1\right) = 0
Solve this equation
The roots of this equation
x1=−97.3893722612836x_{1} = -97.3893722612836
x2=−43.9822971502571x_{2} = -43.9822971502571
x3=−29.845130209103x_{3} = -29.845130209103
x4=56.5486677646163x_{4} = 56.5486677646163
x5=62.2158522491989x_{5} = 62.2158522491989
x6=28.2743338823081x_{6} = 28.2743338823081
x7=29.845130209103x_{7} = 29.845130209103
x8=−81.6814089933346x_{8} = -81.6814089933346
x9=−15.707963267949x_{9} = -15.707963267949
x10=72.2566310325652x_{10} = 72.2566310325652
x11=95.8185759344887x_{11} = 95.8185759344887
x12=45.553093477052x_{12} = 45.553093477052
x13=−55.9326669420193x_{13} = -55.9326669420193
x14=−53.4070751110265x_{14} = -53.4070751110265
x15=18.2335550989418x_{15} = 18.2335550989418
x16=−11.9503697917622x_{16} = -11.9503697917622
x17=100.530964914873x_{17} = 100.530964914873
x18=−9.42477796076938x_{18} = -9.42477796076938
x19=81.6814089933346x_{19} = 81.6814089933346
x20=70.6858347057703x_{20} = 70.6858347057703
x21=−21.9911485751286x_{21} = -21.9911485751286
x22=−14.1371669411541x_{22} = -14.1371669411541
x23=32.0319273584949x_{23} = 32.0319273584949
x24=42.4115008234622x_{24} = 42.4115008234622
x25=−72.2566310325652x_{25} = -72.2566310325652
x26=68.2487278775419x_{26} = 68.2487278775419
x27=15.707963267949x_{27} = 15.707963267949
x28=50.2654824574367x_{28} = 50.2654824574367
x29=−28.2743338823081x_{29} = -28.2743338823081
x30=−91.722187776701x_{30} = -91.722187776701
x31=84.2070008243274x_{31} = 84.2070008243274
x32=−6.28318530717959x_{32} = -6.28318530717959
x33=−17.2787595947439x_{33} = -17.2787595947439
x34=−80.1106126665397x_{34} = -80.1106126665397
x35=−87.9645943005142x_{35} = -87.9645943005142
x36=76.014224508752x_{36} = 76.014224508752
x37=48.6946861306418x_{37} = 48.6946861306418
x38=−20.4203522483337x_{38} = -20.4203522483337
x39=0x_{39} = 0
x40=20.4203522483337x_{40} = 20.4203522483337
x41=89.5353906273091x_{41} = 89.5353906273091
x42=98.0053730838806x_{42} = 98.0053730838806
x43=−65.9734457253857x_{43} = -65.9734457253857
x44=−73.8274273593601x_{44} = -73.8274273593601
x45=−39.8859089924694x_{45} = -39.8859089924694
x46=6.28318530717959x_{46} = 6.28318530717959
x47=−67.5442420521806x_{47} = -67.5442420521806
x48=−36.1283155162826x_{48} = -36.1283155162826
x49=94.2477796076938x_{49} = 94.2477796076938
x50=−61.261056745001x_{50} = -61.261056745001
x51=65.9734457253857x_{51} = 65.9734457253857
x52=36.1283155162826x_{52} = 36.1283155162826
x53=21.9911485751286x_{53} = 21.9911485751286
x54=−33.9415183668907x_{54} = -33.9415183668907
x55=78.5398163397448x_{55} = 78.5398163397448
x56=40.2247036740703x_{56} = 40.2247036740703
x57=−69.7310392015724x_{57} = -69.7310392015724
x58=−47.7398906264439x_{58} = -47.7398906264439
x59=4.71238898038469x_{59} = 4.71238898038469
x60=−3.75759347618678x_{60} = -3.75759347618678
x61=−89.5353906273091x_{61} = -89.5353906273091
x62=7.85398163397448x_{62} = 7.85398163397448
x63=14.1371669411541x_{63} = 14.1371669411541
x64=26.0875367329163x_{64} = 26.0875367329163
x65=86.3937979737193x_{65} = 86.3937979737193
x66=87.9645943005142x_{66} = 87.9645943005142
x67=12.5663706143592x_{67} = 12.5663706143592
x68=−25.7487420513153x_{68} = -25.7487420513153
x69=51.8362787842316x_{69} = 51.8362787842316
x70=−37.6991118430775x_{70} = -37.6991118430775
x71=−51.8362787842316x_{71} = -51.8362787842316
x72=−1.5707963267949x_{72} = -1.5707963267949
x73=59.6902604182061x_{73} = 59.6902604182061
x74=−83.8682061427265x_{74} = -83.8682061427265
x75=−23.5619449019235x_{75} = -23.5619449019235
x76=−99.9149640922764x_{76} = -99.9149640922764
x77=64.4026493985908x_{77} = 64.4026493985908
x78=43.9822971502571x_{78} = 43.9822971502571
x79=−50.2654824574367x_{79} = -50.2654824574367
x80=37.6991118430775x_{80} = 37.6991118430775
x81=−95.8185759344887x_{81} = -95.8185759344887
x82=−64.4026493985908x_{82} = -64.4026493985908
x83=80.1106126665397x_{83} = 80.1106126665397
x84=73.8274273593601x_{84} = 73.8274273593601
x85=−59.6902604182061x_{85} = -59.6902604182061
x86=−7.85398163397448x_{86} = -7.85398163397448
x87=−75.398223686155x_{87} = -75.398223686155
x88=−86.3937979737193x_{88} = -86.3937979737193
x89=23.5619449019235x_{89} = 23.5619449019235
x90=92.6769832808989x_{90} = 92.6769832808989
x91=9.42477796076938x_{91} = 9.42477796076938
x92=54.0230759336235x_{92} = 54.0230759336235
x93=−45.553093477052x_{93} = -45.553093477052
x94=1.5707963267949x_{94} = 1.5707963267949
x95=−58.1194640914112x_{95} = -58.1194640914112
x96=58.1194640914112x_{96} = 58.1194640914112
x97=−94.2477796076938x_{97} = -94.2477796076938
x98=−77.9238155171478x_{98} = -77.9238155171478
x99=−42.4115008234622x_{99} = -42.4115008234622
x100=34.5575191894877x_{100} = 34.5575191894877
x101=−31.4159265358979x_{101} = -31.4159265358979

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
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Convex at the intervals
(−∞,−99.9149640922764]\left(-\infty, -99.9149640922764\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→−∞tan⁡(tan⁡(2x))y = \lim_{x \to -\infty} \tan{\left(\tan{\left(2 x \right)} \right)}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞tan⁡(tan⁡(2x))y = \lim_{x \to \infty} \tan{\left(\tan{\left(2 x \right)} \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(tan(2*x)), 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→−∞(tan⁡(tan⁡(2x))x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(\tan{\left(2 x \right)} \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(tan⁡(2x))x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\tan{\left(2 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:
tan⁡(tan⁡(2x))=−tan⁡(tan⁡(2x))\tan{\left(\tan{\left(2 x \right)} \right)} = - \tan{\left(\tan{\left(2 x \right)} \right)}
- No
tan⁡(tan⁡(2x))=tan⁡(tan⁡(2x))\tan{\left(\tan{\left(2 x \right)} \right)} = \tan{\left(\tan{\left(2 x \right)} \right)}
- Yes
so, the function
is
odd
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