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Plotting a function graph/ tg(3*x)

Graphing y = tg(3*x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = tan(3*x)
f(x)=tan⁡(3x)f{\left(x \right)} = \tan{\left(3 x \right)} 
f = tan(3*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⁡(3x)=0\tan{\left(3 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−55.5014702134197x_{1} = -55.5014702134197
x2=90.0589894029074x_{2} = 90.0589894029074
x3=−33.5103216382911x_{3} = -33.5103216382911
x4=−72.2566310325652x_{4} = -72.2566310325652
x5=−90.0589894029074x_{5} = -90.0589894029074
x6=74.3510261349584x_{6} = 74.3510261349584
x7=−59.6902604182061x_{7} = -59.6902604182061
x8=17.8023583703422x_{8} = 17.8023583703422
x9=−6.28318530717959x_{9} = -6.28318530717959
x10=−85.870199198121x_{10} = -85.870199198121
x11=12.5663706143592x_{11} = 12.5663706143592
x12=56.5486677646163x_{12} = 56.5486677646163
x13=26.1799387799149x_{13} = 26.1799387799149
x14=72.2566310325652x_{14} = 72.2566310325652
x15=37.6991118430775x_{15} = 37.6991118430775
x16=−83.7758040957278x_{16} = -83.7758040957278
x17=92.1533845053006x_{17} = 92.1533845053006
x18=100.530964914873x_{18} = 100.530964914873
x19=48.1710873550435x_{19} = 48.1710873550435
x20=70.162235930172x_{20} = 70.162235930172
x21=94.2477796076938x_{21} = 94.2477796076938
x22=54.4542726622231x_{22} = 54.4542726622231
x23=−24.0855436775217x_{23} = -24.0855436775217
x24=−79.5870138909414x_{24} = -79.5870138909414
x25=80.634211442138x_{25} = 80.634211442138
x26=−87.9645943005142x_{26} = -87.9645943005142
x27=−31.4159265358979x_{27} = -31.4159265358979
x28=46.0766922526503x_{28} = 46.0766922526503
x29=59.6902604182061x_{29} = 59.6902604182061
x30=0x_{30} = 0
x31=30.3687289847013x_{31} = 30.3687289847013
x32=−52.3598775598299x_{32} = -52.3598775598299
x33=50.2654824574367x_{33} = 50.2654824574367
x34=−74.3510261349584x_{34} = -74.3510261349584
x35=−53.4070751110265x_{35} = -53.4070751110265
x36=−92.1533845053006x_{36} = -92.1533845053006
x37=85.870199198121x_{37} = 85.870199198121
x38=−21.9911485751286x_{38} = -21.9911485751286
x39=41.8879020478639x_{39} = 41.8879020478639
x40=32.4631240870945x_{40} = 32.4631240870945
x41=−41.8879020478639x_{41} = -41.8879020478639
x42=63.8790506229925x_{42} = 63.8790506229925
x43=87.9645943005142x_{43} = 87.9645943005142
x44=−19.8967534727354x_{44} = -19.8967534727354
x45=4.18879020478639x_{45} = 4.18879020478639
x46=14.6607657167524x_{46} = 14.6607657167524
x47=−11.5191730631626x_{47} = -11.5191730631626
x48=39.7935069454707x_{48} = 39.7935069454707
x49=−75.398223686155x_{49} = -75.398223686155
x50=81.6814089933346x_{50} = 81.6814089933346
x51=−43.9822971502571x_{51} = -43.9822971502571
x52=−37.6991118430775x_{52} = -37.6991118430775
x53=−65.9734457253857x_{53} = -65.9734457253857
x54=−96.342174710087x_{54} = -96.342174710087
x55=10.471975511966x_{55} = 10.471975511966
x56=−99.4837673636768x_{56} = -99.4837673636768
x57=58.6430628670095x_{57} = 58.6430628670095
x58=76.4454212373516x_{58} = 76.4454212373516
x59=6.28318530717959x_{59} = 6.28318530717959
x60=−39.7935069454707x_{60} = -39.7935069454707
x61=83.7758040957278x_{61} = 83.7758040957278
x62=−17.8023583703422x_{62} = -17.8023583703422
x63=28.2743338823081x_{63} = 28.2743338823081
x64=−70.162235930172x_{64} = -70.162235930172
x65=−68.0678408277789x_{65} = -68.0678408277789
x66=−94.2477796076938x_{66} = -94.2477796076938
x67=−13.6135681655558x_{67} = -13.6135681655558
x68=61.7846555205993x_{68} = 61.7846555205993
x69=43.9822971502571x_{69} = 43.9822971502571
x70=−4.18879020478639x_{70} = -4.18879020478639
x71=−7.33038285837618x_{71} = -7.33038285837618
x72=−9.42477796076938x_{72} = -9.42477796076938
x73=65.9734457253857x_{73} = 65.9734457253857
x74=96.342174710087x_{74} = 96.342174710087
x75=52.3598775598299x_{75} = 52.3598775598299
x76=−30.3687289847013x_{76} = -30.3687289847013
x77=−15.707963267949x_{77} = -15.707963267949
x78=−48.1710873550435x_{78} = -48.1710873550435
x79=−50.2654824574367x_{79} = -50.2654824574367
x80=−46.0766922526503x_{80} = -46.0766922526503
x81=36.6519142918809x_{81} = 36.6519142918809
x82=−35.6047167406843x_{82} = -35.6047167406843
x83=15.707963267949x_{83} = 15.707963267949
x84=−57.5958653158129x_{84} = -57.5958653158129
x85=−77.4926187885482x_{85} = -77.4926187885482
x86=24.0855436775217x_{86} = 24.0855436775217
x87=−61.7846555205993x_{87} = -61.7846555205993
x88=21.9911485751286x_{88} = 21.9911485751286
x89=34.5575191894877x_{89} = 34.5575191894877
x90=−26.1799387799149x_{90} = -26.1799387799149
x91=−97.3893722612836x_{91} = -97.3893722612836
x92=−2.0943951023932x_{92} = -2.0943951023932
x93=−28.2743338823081x_{93} = -28.2743338823081
x94=78.5398163397448x_{94} = 78.5398163397448
x95=98.4365698124802x_{95} = 98.4365698124802
x96=−63.8790506229925x_{96} = -63.8790506229925
x97=2.0943951023932x_{97} = 2.0943951023932
x98=68.0678408277789x_{98} = 68.0678408277789
x99=19.8967534727354x_{99} = 19.8967534727354
x100=−81.6814089933346x_{100} = -81.6814089933346
x101=8.37758040957278x_{101} = 8.37758040957278
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(3*x).
tan⁡(0⋅3)\tan{\left(0 \cdot 3 \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
3tan⁡2(3x)+3=03 \tan^{2}{\left(3 x \right)} + 3 = 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
18(tan⁡2(3x)+1)tan⁡(3x)=018 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0

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
[0,∞)\left[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞tan⁡(3x)=⟨−∞,∞⟩\lim_{x \to -\infty} \tan{\left(3 x \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→∞tan⁡(3x)=⟨−∞,∞⟩\lim_{x \to \infty} \tan{\left(3 x \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 tan(3*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⁡(3x)x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(3x)x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \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⁡(3x)=−tan⁡(3x)\tan{\left(3 x \right)} = - \tan{\left(3 x \right)}
- No
tan⁡(3x)=tan⁡(3x)\tan{\left(3 x \right)} = \tan{\left(3 x \right)}
- Yes
so, the function
is
odd
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