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Trazado el gráfico de la función/ tg(tg(tg(tgx)))-1

Gráfico de la función y = tg(tg(tg(tgx)))-1

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = tan(tan(tan(tan(x)))) - 1
f(x)=tan⁡(tan⁡(tan⁡(tan⁡(x))))−1f{\left(x \right)} = \tan{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} - 1 
f = tan(tan(tan(tan(x)))) - 1
Gráfico de la función
02468-8-6-4-2-1010-2525
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
tan⁡(tan⁡(tan⁡(tan⁡(x))))−1=0\tan{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} - 1 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=atan⁡(atan⁡(atan⁡(π4)))x_{1} = \operatorname{atan}{\left(\operatorname{atan}{\left(\operatorname{atan}{\left(\frac{\pi}{4} \right)} \right)} \right)}
Solución numérica
x1=−63.7025185492294x_{1} = -63.7025185492294
x2=86.2492645429153x_{2} = 86.2492645429153
x3=75.9293151963108x_{3} = 75.9293151963108
x4=−37.1680203329218x_{4} = -37.1680203329218
x5=−49.7343909472809x_{5} = -49.7343909472809
x6=−68.5839468688197x_{6} = -68.5839468688197
x7=−25.8449266329477x_{7} = -25.8449266329477
x8=−86.0206312958681x_{8} = -86.0206312958681
x9=−90.5750954439481x_{9} = -90.5750954439481
x10=−92.0549129210679x_{10} = -92.0549129210679
x11=101.451216293452x_{11} = 101.451216293452
x12=96.5187067838501x_{12} = 96.5187067838501
x13=90.2355214766705x_{13} = 90.2355214766705
x14=64.2581159677868x_{14} = 64.2581159677868
x15=42.8349654400481x_{15} = 42.8349654400481
x16=19.3806474316945x_{16} = 19.3806474316945
x17=−27.7432423721524x_{17} = -27.7432423721524
x18=−43.4512056401013x_{18} = -43.4512056401013
x19=−79.7374459886885x_{19} = -79.7374459886885
x20=−3.92414864765278x_{20} = -3.92414864765278
x21=−71.7255395224095x_{21} = -71.7255395224095
x22=−12.0352791042034x_{22} = -12.0352791042034
x23=−35.7551488384314x_{23} = -35.7551488384314
x24=60.2213519283618x_{24} = 60.2213519283618
x25=−17.7516411791057x_{25} = -17.7516411791057
x26=−43.1486285828937x_{26} = -43.1486285828937
x27=−99.9998734047176x_{27} = -99.9998734047176
x28=−96.0805813981521x_{28} = -96.0805813981521
x29=−78.0087248295891x_{29} = -78.0087248295891
x30=98.3096236398618x_{30} = 98.3096236398618
x31=48.4326806669784x_{31} = 48.4326806669784
x32=82.2125005034904x_{32} = 82.2125005034904
x33=94.7788711178496x_{33} = 94.7788711178496
x34=24.2620757512848x_{34} = 24.2620757512848
x35=46.2532243264134x_{35} = 46.2532243264134
x36=14.2498431959961x_{36} = 14.2498431959961
x37=75.9293151963108x_{37} = 75.9293151963108
x38=−5.75209379702383x_{38} = -5.75209379702383
x39=−55.8032941633785x_{39} = -55.8032941633785
x40=16.2390547781047x_{40} = 16.2390547781047
x41=91.9398555214674x_{41} = 91.9398555214674
x42=−13.7640002633028x_{42} = -13.7640002633028
x43=−0.712185404229327x_{43} = -0.712185404229327
x44=42.7846675013135x_{44} = 42.7846675013135
x45=−2.04367791115677x_{45} = -2.04367791115677
x46=0.531091510155762x_{46} = 0.531091510155762
x47=53.9381666211822x_{47} = 53.9381666211822
x48=−57.7462974135599x_{48} = -57.7462974135599
x49=−46.5927982936911x_{49} = -46.5927982936911
x50=38.2302033532333x_{50} = 38.2302033532333
x51=−93.716688097538x_{51} = -93.716688097538
x52=−709.468848201137x_{52} = -709.468848201137
x53=9.95586947092514x_{53} = 9.95586947092514
x54=4.06184403216802x_{54} = 4.06184403216802
x55=28.8054253924639x_{55} = 28.8054253924639
x56=−51.8727443194123x_{56} = -51.8727443194123
x57=32.2495951032613x_{57} = 32.2495951032613
x58=57.9986642593673x_{58} = 57.9986642593673
x59=7.93726803943923x_{59} = 7.93726803943923
x60=−34.026427679332x_{60} = -34.026427679332
x61=−59.1591689080503x_{61} = -59.1591689080503
x62=70.2129531214085x_{62} = 70.2129531214085
x63=−14.3389424380385x_{63} = -14.3389424380385
x64=41.7609558752455x_{64} = 41.7609558752455
x65=−61.7339383293628x_{65} = -61.7339383293628
x66=−23.7064783327274x_{66} = -23.7064783327274
x67=−7.92124445012173x_{67} = -7.92124445012173
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en tan(tan(tan(tan(x)))) - 1.
−1+tan⁡(tan⁡(tan⁡(tan⁡(0))))-1 + \tan{\left(\tan{\left(\tan{\left(\tan{\left(0 \right)} \right)} \right)} \right)}
Resultado:
f(0)=−1f{\left(0 \right)} = -1
Punto:
(0, -1)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
(tan⁡2(x)+1)(tan⁡2(tan⁡(x))+1)(tan⁡2(tan⁡(tan⁡(x)))+1)(tan⁡2(tan⁡(tan⁡(tan⁡(x))))+1)=0\left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(x \right)} \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} + 1\right) = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(tan⁡(tan⁡(tan⁡(tan⁡(x))))−1)y = \lim_{x \to -\infty}\left(\tan{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} - 1\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(tan⁡(tan⁡(tan⁡(x))))−1)y = \lim_{x \to \infty}\left(\tan{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} - 1\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(tan(tan(tan(x)))) - 1, dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(tan⁡(tan⁡(tan⁡(tan⁡(x))))−1x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} - 1}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(tan⁡(tan⁡(tan⁡(tan⁡(x))))−1x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} - 1}{x}\right)
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
tan⁡(tan⁡(tan⁡(tan⁡(x))))−1=−tan⁡(tan⁡(tan⁡(tan⁡(x))))−1\tan{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} - 1 = - \tan{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} - 1
- No
tan⁡(tan⁡(tan⁡(tan⁡(x))))−1=tan⁡(tan⁡(tan⁡(tan⁡(x))))+1\tan{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} - 1 = \tan{\left(\tan{\left(\tan{\left(\tan{\left(x \right)} \right)} \right)} \right)} + 1
- No
es decir, función
no es
par ni impar
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