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Plotting a function graph/ cos(7x)+cos(3x)

Graphing y = cos(7x)+cos(3x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cos(7*x) + cos(3*x)
f(x)=cos⁡(3x)+cos⁡(7x)f{\left(x \right)} = \cos{\left(3 x \right)} + \cos{\left(7 x \right)} 
f = cos(3*x) + cos(7*x)
The graph of the function
02468-8-6-4-2-10105-5
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:
cos⁡(3x)+cos⁡(7x)=0\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−9π10x_{1} = - \frac{9 \pi}{10}
x2=−3π4x_{2} = - \frac{3 \pi}{4}
x3=−π2x_{3} = - \frac{\pi}{2}
x4=−π4x_{4} = - \frac{\pi}{4}
x5=π10x_{5} = \frac{\pi}{10}
x6=π4x_{6} = \frac{\pi}{4}
x7=π2x_{7} = \frac{\pi}{2}
x8=3π4x_{8} = \frac{3 \pi}{4}
x9=−ilog⁡(−105+58+25+58+i4+5i4)x_{9} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x10=−ilog⁡(−25+58+105+58−5i4−i4)x_{10} = - i \log{\left(- \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{8} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{8} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
x11=−ilog⁡(−105−516+25−516+25+516+105+516+i4+5i4)x_{11} = - i \log{\left(- \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} + \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x12=−ilog⁡(−105+516−105−516−25+516+25−516−i4+5i4)x_{12} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} - \frac{i}{4} + \frac{\sqrt{5} i}{4} \right)}
x13=−ilog⁡(−105+516−25+516−25−516+105−516−5i4−i4)x_{13} = - i \log{\left(- \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} - \frac{\sqrt{5} i}{4} - \frac{i}{4} \right)}
x14=−ilog⁡(−25−516+25+516+105−516+105+516−5i4+i4)x_{14} = - i \log{\left(- \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{2} \sqrt{\sqrt{5} + 5}}{16} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{16} + \frac{\sqrt{10} \sqrt{\sqrt{5} + 5}}{16} - \frac{\sqrt{5} i}{4} + \frac{i}{4} \right)}
Numerical solution
x1=−9.73893722612836x_{1} = -9.73893722612836
x2=−0.785398163397448x_{2} = -0.785398163397448
x3=−87.6504350351552x_{3} = -87.6504350351552
x4=46.18141200777x_{4} = 46.18141200777
x5=56.2345084992573x_{5} = 56.2345084992573
x6=26.0752190247953x_{6} = 26.0752190247953
x7=84.037603483527x_{7} = 84.037603483527
x8=54.1924732744239x_{8} = 54.1924732744239
x9=−68.329640215578x_{9} = -68.329640215578
x10=93.9336203423348x_{10} = 93.9336203423348
x11=19.6349540849362x_{11} = 19.6349540849362
x12=66.2876049907446x_{12} = 66.2876049907446
x13=92.0486647501809x_{13} = 92.0486647501809
x14=33.6150413934108x_{14} = 33.6150413934108
x15=−5.96902604182061x_{15} = -5.96902604182061
x16=−41.7831822927443x_{16} = -41.7831822927443
x17=−60.0044196835651x_{17} = -60.0044196835651
x18=−51.8362787842316x_{18} = -51.8362787842316
x19=−80.8960108299372x_{19} = -80.8960108299372
x20=88.2787535658732x_{20} = 88.2787535658732
x21=−47.9092879672443x_{21} = -47.9092879672443
x22=−49.9513231920777x_{22} = -49.9513231920777
x23=58.1194640914112x_{23} = 58.1194640914112
x24=60.0044196835651x_{24} = 60.0044196835651
x25=−16.6504410640259x_{25} = -16.6504410640259
x26=−49.3230046613598x_{26} = -49.3230046613598
x27=38.0132711084365x_{27} = 38.0132711084365
x28=−69.9004365423729x_{28} = -69.9004365423729
x29=4.08407044966673x_{29} = 4.08407044966673
x30=95.1902574037707x_{30} = 95.1902574037707
x31=−3.92699081698724x_{31} = -3.92699081698724
x32=−55.7632696012188x_{32} = -55.7632696012188
x33=−95.8185759344887x_{33} = -95.8185759344887
x34=−17.9070781254618x_{34} = -17.9070781254618
x35=62.0464549083984x_{35} = 62.0464549083984
x36=−93.9336203423348x_{36} = -93.9336203423348
x37=22.3053078404875x_{37} = 22.3053078404875
x38=80.1106126665397x_{38} = 80.1106126665397
x39=16.0221225333079x_{39} = 16.0221225333079
x40=−31.7300858012569x_{40} = -31.7300858012569
x41=12.2522113490002x_{41} = 12.2522113490002
x42=−77.7544181763474x_{42} = -77.7544181763474
x43=81.9955682586936x_{43} = 81.9955682586936
x44=91.8915851175014x_{44} = 91.8915851175014
x45=−29.845130209103x_{45} = -29.845130209103
x46=70.0575161750524x_{46} = 70.0575161750524
x47=−61.8893752757189x_{47} = -61.8893752757189
x48=−53.7212343763855x_{48} = -53.7212343763855
x49=−81.9955682586936x_{49} = -81.9955682586936
x50=58.7477826221291x_{50} = 58.7477826221291
x51=44.2964564156161x_{51} = 44.2964564156161
x52=38.484510006475x_{52} = 38.484510006475
x53=48.0663675999238x_{53} = 48.0663675999238
x54=−39.8982267005904x_{54} = -39.8982267005904
x55=−58.9048622548086x_{55} = -58.9048622548086
x56=2.19911485751286x_{56} = 2.19911485751286
x57=40.0553063332699x_{57} = 40.0553063332699
x58=−85.7654794430014x_{58} = -85.7654794430014
x59=49.9513231920777x_{59} = 49.9513231920777
x60=−43.6681378848981x_{60} = -43.6681378848981
x61=−71.9424717672063x_{61} = -71.9424717672063
x62=14.1371669411541x_{62} = 14.1371669411541
x63=98.174770424681x_{63} = 98.174770424681
x64=10.2101761241668x_{64} = 10.2101761241668
x65=100.216805649514x_{65} = 100.216805649514
x66=−16.0221225333079x_{66} = -16.0221225333079
x67=−90.3207887907066x_{67} = -90.3207887907066
x68=0.314159265358979x_{68} = 0.314159265358979
x69=90.1637091580271x_{69} = 90.1637091580271
x70=71.9424717672063x_{70} = 71.9424717672063
x71=−21.6769893097696x_{71} = -21.6769893097696
x72=78.2256570743859x_{72} = 78.2256570743859
x73=36.1283155162826x_{73} = 36.1283155162826
x74=74.6128255227576x_{74} = 74.6128255227576
x75=−99.7455667514759x_{75} = -99.7455667514759
x76=−33.7721210260903x_{76} = -33.7721210260903
x77=76.1836218495525x_{77} = 76.1836218495525
x78=18.0641577581413x_{78} = 18.0641577581413
x79=−36.9137136796801x_{79} = -36.9137136796801
x80=27.9601746169492x_{80} = 27.9601746169492
x81=162.577419823272x_{81} = 162.577419823272
x82=24.1902634326414x_{82} = 24.1902634326414
x83=−19.7920337176157x_{83} = -19.7920337176157
x84=68.1725605828985x_{84} = 68.1725605828985
x85=−75.712382951514x_{85} = -75.712382951514
x86=−63.7743308678728x_{86} = -63.7743308678728
x87=−11.7809724509617x_{87} = -11.7809724509617
x88=−7.85398163397448x_{88} = -7.85398163397448
x89=5.96902604182061x_{89} = 5.96902604182061
x90=−38.0132711084365x_{90} = -38.0132711084365
x91=−91.8915851175014x_{91} = -91.8915851175014
x92=−73.8274273593601x_{92} = -73.8274273593601
x93=32.2013246992954x_{93} = 32.2013246992954
x94=−97.7035315266426x_{94} = -97.7035315266426
x95=34.2433599241287x_{95} = 34.2433599241287
x96=−27.9601746169492x_{96} = -27.9601746169492
x97=−25.9181393921158x_{97} = -25.9181393921158
x98=−65.6592864600267x_{98} = -65.6592864600267
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(7*x) + cos(3*x).
cos⁡(0⋅7)+cos⁡(0⋅3)\cos{\left(0 \cdot 7 \right)} + \cos{\left(0 \cdot 3 \right)}
The result:
f(0)=2f{\left(0 \right)} = 2
The point:
(0, 2)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(cos⁡(3x)+cos⁡(7x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\cos{\left(3 x \right)} + \cos{\left(7 x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(cos⁡(3x)+cos⁡(7x))=⟨−2,2⟩\lim_{x \to \infty}\left(\cos{\left(3 x \right)} + \cos{\left(7 x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(7*x) + cos(3*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(3x)+cos⁡(7x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(3 x \right)} + \cos{\left(7 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(3x)+cos⁡(7x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(3 x \right)} + \cos{\left(7 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
cos⁡(3x)+cos⁡(7x)=cos⁡(3x)+cos⁡(7x)\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = \cos{\left(3 x \right)} + \cos{\left(7 x \right)}
- Yes
cos⁡(3x)+cos⁡(7x)=−cos⁡(3x)−cos⁡(7x)\cos{\left(3 x \right)} + \cos{\left(7 x \right)} = - \cos{\left(3 x \right)} - \cos{\left(7 x \right)}
- No
so, the function
is
even
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