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tg(4x)
Plotting a function graph/ tg(4x)

Graphing y = tg(4x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = tan(4*x)
f(x)=tan⁡(4x)f{\left(x \right)} = \tan{\left(4 x \right)} 
f = tan(4*x)
The graph of the function
0-10102030405060708090-100100
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⁡(4x)=0\tan{\left(4 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−11.7809724509617x_{1} = -11.7809724509617
x2=90.3207887907066x_{2} = 90.3207887907066
x3=18.0641577581413x_{3} = 18.0641577581413
x4=40.0553063332699x_{4} = 40.0553063332699
x5=29.845130209103x_{5} = 29.845130209103
x6=28.2743338823081x_{6} = 28.2743338823081
x7=−93.4623814442964x_{7} = -93.4623814442964
x8=60.4756585816035x_{8} = 60.4756585816035
x9=46.3384916404494x_{9} = 46.3384916404494
x10=33.7721210260903x_{10} = 33.7721210260903
x11=91.8915851175014x_{11} = 91.8915851175014
x12=−43.9822971502571x_{12} = -43.9822971502571
x13=−63.6172512351933x_{13} = -63.6172512351933
x14=−99.7455667514759x_{14} = -99.7455667514759
x15=3.92699081698724x_{15} = 3.92699081698724
x16=−7.85398163397448x_{16} = -7.85398163397448
x17=−19.6349540849362x_{17} = -19.6349540849362
x18=36.1283155162826x_{18} = 36.1283155162826
x19=82.4668071567321x_{19} = 82.4668071567321
x20=−21.9911485751286x_{20} = -21.9911485751286
x21=6.28318530717959x_{21} = 6.28318530717959
x22=−40.0553063332699x_{22} = -40.0553063332699
x23=−33.7721210260903x_{23} = -33.7721210260903
x24=−32.2013246992954x_{24} = -32.2013246992954
x25=32.2013246992954x_{25} = 32.2013246992954
x26=−71.4712328691678x_{26} = -71.4712328691678
x27=−3.92699081698724x_{27} = -3.92699081698724
x28=84.037603483527x_{28} = 84.037603483527
x29=42.4115008234622x_{29} = 42.4115008234622
x30=68.329640215578x_{30} = 68.329640215578
x31=−29.845130209103x_{31} = -29.845130209103
x32=−14.1371669411541x_{32} = -14.1371669411541
x33=−69.9004365423729x_{33} = -69.9004365423729
x34=−67.5442420521806x_{34} = -67.5442420521806
x35=−59.6902604182061x_{35} = -59.6902604182061
x36=11.7809724509617x_{36} = 11.7809724509617
x37=−80.1106126665397x_{37} = -80.1106126665397
x38=2.35619449019234x_{38} = 2.35619449019234
x39=−15.707963267949x_{39} = -15.707963267949
x40=−41.6261026600648x_{40} = -41.6261026600648
x41=98.174770424681x_{41} = 98.174770424681
x42=80.1106126665397x_{42} = 80.1106126665397
x43=55.7632696012188x_{43} = 55.7632696012188
x44=16.4933614313464x_{44} = 16.4933614313464
x45=95.8185759344887x_{45} = 95.8185759344887
x46=−62.0464549083984x_{46} = -62.0464549083984
x47=−47.9092879672443x_{47} = -47.9092879672443
x48=−89.5353906273091x_{48} = -89.5353906273091
x49=54.1924732744239x_{49} = 54.1924732744239
x50=65.9734457253857x_{50} = 65.9734457253857
x51=10.2101761241668x_{51} = 10.2101761241668
x52=69.9004365423729x_{52} = 69.9004365423729
x53=−98.174770424681x_{53} = -98.174770424681
x54=72.2566310325652x_{54} = 72.2566310325652
x55=47.9092879672443x_{55} = 47.9092879672443
x56=−37.6991118430775x_{56} = -37.6991118430775
x57=94.2477796076938x_{57} = 94.2477796076938
x58=50.2654824574367x_{58} = 50.2654824574367
x59=−95.8185759344887x_{59} = -95.8185759344887
x60=43.9822971502571x_{60} = 43.9822971502571
x61=62.0464549083984x_{61} = 62.0464549083984
x62=7.85398163397448x_{62} = 7.85398163397448
x63=−10.2101761241668x_{63} = -10.2101761241668
x64=−65.9734457253857x_{64} = -65.9734457253857
x65=87.9645943005142x_{65} = 87.9645943005142
x66=24.3473430653209x_{66} = 24.3473430653209
x67=76.1836218495525x_{67} = 76.1836218495525
x68=−87.9645943005142x_{68} = -87.9645943005142
x69=−25.9181393921158x_{69} = -25.9181393921158
x70=−77.7544181763474x_{70} = -77.7544181763474
x71=25.9181393921158x_{71} = 25.9181393921158
x72=−76.1836218495525x_{72} = -76.1836218495525
x73=−85.6083998103219x_{73} = -85.6083998103219
x74=78.5398163397448x_{74} = 78.5398163397448
x75=−54.1924732744239x_{75} = -54.1924732744239
x76=0x_{76} = 0
x77=20.4203522483337x_{77} = 20.4203522483337
x78=−5.49778714378214x_{78} = -5.49778714378214
x79=−27.4889357189107x_{79} = -27.4889357189107
x80=51.8362787842316x_{80} = 51.8362787842316
x81=100.530964914873x_{81} = 100.530964914873
x82=−36.1283155162826x_{82} = -36.1283155162826
x83=−51.8362787842316x_{83} = -51.8362787842316
x84=64.4026493985908x_{84} = 64.4026493985908
x85=21.9911485751286x_{85} = 21.9911485751286
x86=38.484510006475x_{86} = 38.484510006475
x87=58.1194640914112x_{87} = 58.1194640914112
x88=−58.1194640914112x_{88} = -58.1194640914112
x89=73.8274273593601x_{89} = 73.8274273593601
x90=−81.6814089933346x_{90} = -81.6814089933346
x91=−73.8274273593601x_{91} = -73.8274273593601
x92=−18.0641577581413x_{92} = -18.0641577581413
x93=−23.5619449019235x_{93} = -23.5619449019235
x94=−84.037603483527x_{94} = -84.037603483527
x95=14.1371669411541x_{95} = 14.1371669411541
x96=−91.8915851175014x_{96} = -91.8915851175014
x97=−1.5707963267949x_{97} = -1.5707963267949
x98=−49.4800842940392x_{98} = -49.4800842940392
x99=−45.553093477052x_{99} = -45.553093477052
x100=−55.7632696012188x_{100} = -55.7632696012188
x101=86.3937979737193x_{101} = 86.3937979737193
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(4*x).
tan⁡(4⋅0)\tan{\left(4 \cdot 0 \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
4tan⁡2(4x)+4=04 \tan^{2}{\left(4 x \right)} + 4 = 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
32(tan⁡2(4x)+1)tan⁡(4x)=032 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 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⁡(4x)=⟨−∞,∞⟩\lim_{x \to -\infty} \tan{\left(4 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⁡(4x)=⟨−∞,∞⟩\lim_{x \to \infty} \tan{\left(4 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(4*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(tan⁡(4x)x)=lim⁡x→−∞(tan⁡(4x)x)\lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(tan⁡(4x)x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right)
lim⁡x→∞(tan⁡(4x)x)=lim⁡x→∞(tan⁡(4x)x)\lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(4x)x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(4 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⁡(4x)=−tan⁡(4x)\tan{\left(4 x \right)} = - \tan{\left(4 x \right)}
- No
tan⁡(4x)=tan⁡(4x)\tan{\left(4 x \right)} = \tan{\left(4 x \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = tg(4x)
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