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Plotting a function graph/ -sin(x)+(-x+tan(x))/x

Graphing y = -sin(x)+(-x+tan(x))/x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                 -x + tan(x)
f(x) = -sin(x) + -----------
                      x
f(x)=−sin⁡(x)+−x+tan⁡(x)xf{\left(x \right)} = - \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x} 
f = -sin(x) + (-x + tan(x))/x
The graph of the function
02468-8-6-4-2-1010-50100
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
−sin⁡(x)+−x+tan⁡(x)x=0- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=−101.833606800387x_{1} = -101.833606800387
x2=−7.22618054111001x_{2} = -7.22618054111001
x3=−76.6746429883172x_{3} = -76.6746429883172
x4=−2401.65357112653x_{4} = -2401.65357112653
x5=−70.3830689311537x_{5} = -70.3830689311537
x6=29.4428576232766x_{6} = 29.4428576232766
x7=−32.5975001637956x_{7} = -32.5975001637956
x8=2920.02228019732x_{8} = 2920.02228019732
x9=67.2369014769033x_{9} = 67.2369014769033
x10=−19.9644318693588x_{10} = -19.9644318693588
x11=54.6489448261668x_{11} = 54.6489448261668
x12=−64.0904427123236x_{12} = -64.0904427123236
x13=−45.2031364418958x_{13} = -45.2031364418958
x14=4666.76052840282x_{14} = 4666.76052840282
x15=1864.43296213262x_{15} = 1864.43296213262
x16=92.4001120712934x_{16} = 92.4001120712934
x17=−57.7965084032003x_{17} = -57.7965084032003
x18=−89.2553484311087x_{18} = -89.2553484311087
x19=16.7969368070305x_{19} = 16.7969368070305
x20=79.8200967348283x_{20} = 79.8200967348283
x21=10.4354574274462x_{21} = 10.4354574274462
x22=−13.6221601570929x_{22} = -13.6221601570929
x23=86.1104336766229x_{23} = 86.1104336766229
x24=35.7505852973437x_{24} = 35.7505852973437
x25=186.705219429264x_{25} = 186.705219429264
x26=60.9436584385652x_{26} = 60.9436584385652
x27=98.6892313840924x_{27} = 98.6892313840924
x28=73.5289743541661x_{28} = 73.5289743541661
x29=−26.2862433755271x_{29} = -26.2862433755271
x30=104.977870897937x_{30} = 104.977870897937
x31=−51.5009104568073x_{31} = -51.5009104568073
x32=−38.9024128902686x_{32} = -38.9024128902686
x33=48.3523359733182x_{33} = 48.3523359733182
x34=42.053206401595x_{34} = 42.053206401595
x35=−16984.9716772703x_{35} = -16984.9716772703
x36=3.96014663739241x_{36} = 3.96014663739241
x37=23.1270642155098x_{37} = 23.1270642155098
x38=−95.5447362521662x_{38} = -95.5447362521662
x39=−82.9653547073319x_{39} = -82.9653547073319
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to -sin(x) + (-x + tan(x))/x.
−sin⁡(0)+−0+tan⁡(0)0- \sin{\left(0 \right)} + \frac{- 0 + \tan{\left(0 \right)}}{0}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
−cos⁡(x)+tan⁡2(x)x−−x+tan⁡(x)x2=0- \cos{\left(x \right)} + \frac{\tan^{2}{\left(x \right)}}{x} - \frac{- x + \tan{\left(x \right)}}{x^{2}} = 0
Solve this equation
The roots of this equation
x1=13.7103766513378x_{1} = 13.7103766513378
x2=45.2692689800094x_{2} = 45.2692689800094
x3=99.1780792159165x_{3} = 99.1780792159165
x4=83.0211302986392x_{4} = 83.0211302986392
x5=20.0456166959086x_{5} = 20.0456166959086
x6=76.731703756933x_{6} = 76.731703756933
x7=−48.4172721563853x_{7} = -48.4172721563853
x8=32.6695676392345x_{8} = 32.6695676392345
x9=51.5647356400495x_{9} = 51.5647356400495
x10=80.3447087346139x_{10} = 80.3447087346139
x11=−92.4541638982143x_{11} = -92.4541638982143
x12=−10.5283877748772x_{12} = -10.5283877748772
x13=−14.5629043518023x_{13} = -14.5629043518023
x14=17.675544751362x_{14} = 17.675544751362
x15=61.5175136492539x_{15} = 61.5175136492539
x16=1493.91489693543x_{16} = 1493.91489693543
x17=−70.930100123665x_{17} = -70.930100123665
x18=23.9180746722454x_{18} = 23.9180746722454
x19=−83.483265017145x_{19} = -83.483265017145
x20=−96.038880810861x_{20} = -96.038880810861
x21=−86.16560960098x_{21} = -86.16560960098
x22=−67.2961450984475x_{22} = -67.2961450984475
x23=48.9720455511274x_{23} = 48.9720455511274
x24=−98.7422477693976x_{24} = -98.7422477693976
x25=64.1504954681049x_{25} = 64.1504954681049
x26=11.4608266550795x_{26} = 11.4608266550795
x27=−79.8764998162761x_{27} = -79.8764998162761
x28=−27.04454788015x_{28} = -27.04454788015
x29=38.971246687766x_{29} = 38.971246687766
x30=−54.7117340971332x_{30} = -54.7117340971332
x31=−29.516806008253x_{31} = -29.516806008253
x32=89.3099503974961x_{32} = 89.3099503974961
x33=74.0681086793484x_{33} = 74.0681086793484
x34=−20.7946535700566x_{34} = -20.7946535700566
x35=−33.3037402798478x_{35} = -33.3037402798478
x36=−35.8209580945664x_{36} = -35.8209580945664
x37=−52.1077748967443x_{37} = -52.1077748967443
x38=70.4415467276822x_{38} = 70.4415467276822
x39=−8.37839374690101x_{39} = -8.37839374690101
x40=30.1732794211017x_{40} = 30.1732794211017
x41=−89.7608179516488x_{41} = -89.7608179516488
x42=92.8997907669398x_{42} = 92.8997907669398
x43=−2.47351749621617x_{43} = -2.47351749621617
x44=−39.568478688083x_{44} = -39.568478688083
x45=−64.6547752099016x_{45} = -64.6547752099016
x46=−58.3805640516404x_{46} = -58.3805640516404
x47=55.2439678724582x_{47} = 55.2439678724582
x48=−61.0045681868314x_{48} = -61.0045681868314
x49=−4.06796070202516x_{49} = -4.06796070202516
x50=67.7923138841664x_{50} = 67.7923138841664
x51=−45.8368540585271x_{51} = -45.8368540585271
x52=−77.2063178228991x_{52} = -77.2063178228991
x53=−42.1206330735756x_{53} = -42.1206330735756
x54=26.3622989497276x_{54} = 26.3622989497276
x55=42.7022928266836x_{55} = 42.7022928266836
The values of the extrema at the points:
(13.710376651337784, -1.74990601464325)

(45.26926898000935, -1.88426276933223)

(99.1780792159165, -0.0691846821658562)

(83.02113029863915, -1.92222537226944)

(20.0456166959086, -1.80377116506546)

(76.73170375693304, -1.9180913909594)

(-48.4172721563853, -1.8892355015881)

(32.669567639234465, -1.85687158585879)

(51.56473564004946, -1.89370407662581)

(80.34470873461386, -0.0794685884508887)

(-92.45416389821432, -1.92754199787997)

(-10.528387774877206, -1.70454518326009)

(-14.562904351802331, -0.240691971476041)

(17.675544751361993, -0.212713684311768)

(61.517513649253935, -0.0946945315073272)

(1493.9148969354255, -0.011456219865184)

(-70.93010012366503, -0.0862497416241246)

(23.918074672245385, -0.17514043834615)

(-83.483265017145, -0.0774912179827233)

(-96.03888081086099, -0.0706658235072578)

(-86.16560960097998, -1.92410408680473)

(-67.29614509844748, -1.91072074380383)

(48.97204555112736, -0.109942751950698)

(-98.74224776939764, -1.93061433472546)

(64.15049546810489, -1.90787234572801)

(11.460826655079458, -0.280101245395084)

(-79.87649981627605, -1.92022548530831)

(-27.044547880150045, -0.161777548023469)

(38.97124668776598, -1.87238457707619)

(-54.71173409713315, -1.89774577888521)

(-29.51680600825305, -1.84713107877411)

(89.30995039749605, -1.92587298238211)

(74.06810867934836, -0.0838321279683099)

(-20.79465357005665, -0.191657401395929)

(-33.303740279847794, -0.141352241571297)

(-35.82095809456636, -1.86518682472331)

(-52.10777489674433, -0.105569960666872)

(70.44154672768221, -1.91335796606589)

(-8.378393746901006, -0.340722060743889)

(30.173279421101697, -0.150704537140616)

(-89.76081795164879, -0.0738817739672398)

(92.89979076693976, -0.072229011694115)

(-2.47351749621617, -0.699553755196199)

(-39.56847868808304, -0.126357219324199)

(-64.6547752099016, -0.0916557142382977)

(-58.38056405164039, -0.0979989547585743)

(55.24396787245825, -0.101608210744567)

(-61.00456818683137, -1.90478427696177)

(-4.0679607020251645, -1.47231112804526)

(67.7923138841664, -0.0888498984460918)

(-45.836854058527116, -0.114799392692209)

(-77.20631782289905, -0.0815773704077247)

(-42.120633073575554, -1.87868789765566)

(26.36229894972762, -1.83552691884256)

(42.70229282668361, -0.120231686165023)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=13.7103766513378x_{1} = 13.7103766513378
x2=45.2692689800094x_{2} = 45.2692689800094
x3=83.0211302986392x_{3} = 83.0211302986392
x4=20.0456166959086x_{4} = 20.0456166959086
x5=76.731703756933x_{5} = 76.731703756933
x6=−48.4172721563853x_{6} = -48.4172721563853
x7=32.6695676392345x_{7} = 32.6695676392345
x8=51.5647356400495x_{8} = 51.5647356400495
x9=−92.4541638982143x_{9} = -92.4541638982143
x10=−10.5283877748772x_{10} = -10.5283877748772
x11=−86.16560960098x_{11} = -86.16560960098
x12=−67.2961450984475x_{12} = -67.2961450984475
x13=−98.7422477693976x_{13} = -98.7422477693976
x14=64.1504954681049x_{14} = 64.1504954681049
x15=−79.8764998162761x_{15} = -79.8764998162761
x16=38.971246687766x_{16} = 38.971246687766
x17=−54.7117340971332x_{17} = -54.7117340971332
x18=−29.516806008253x_{18} = -29.516806008253
x19=89.3099503974961x_{19} = 89.3099503974961
x20=−35.8209580945664x_{20} = -35.8209580945664
x21=70.4415467276822x_{21} = 70.4415467276822
x22=−61.0045681868314x_{22} = -61.0045681868314
x23=−4.06796070202516x_{23} = -4.06796070202516
x24=−42.1206330735756x_{24} = -42.1206330735756
x25=26.3622989497276x_{25} = 26.3622989497276
Maxima of the function at points:
x25=99.1780792159165x_{25} = 99.1780792159165
x25=80.3447087346139x_{25} = 80.3447087346139
x25=−14.5629043518023x_{25} = -14.5629043518023
x25=17.675544751362x_{25} = 17.675544751362
x25=61.5175136492539x_{25} = 61.5175136492539
x25=1493.91489693543x_{25} = 1493.91489693543
x25=−70.930100123665x_{25} = -70.930100123665
x25=23.9180746722454x_{25} = 23.9180746722454
x25=−83.483265017145x_{25} = -83.483265017145
x25=−96.038880810861x_{25} = -96.038880810861
x25=48.9720455511274x_{25} = 48.9720455511274
x25=11.4608266550795x_{25} = 11.4608266550795
x25=−27.04454788015x_{25} = -27.04454788015
x25=74.0681086793484x_{25} = 74.0681086793484
x25=−20.7946535700566x_{25} = -20.7946535700566
x25=−33.3037402798478x_{25} = -33.3037402798478
x25=−52.1077748967443x_{25} = -52.1077748967443
x25=−8.37839374690101x_{25} = -8.37839374690101
x25=30.1732794211017x_{25} = 30.1732794211017
x25=−89.7608179516488x_{25} = -89.7608179516488
x25=92.8997907669398x_{25} = 92.8997907669398
x25=−2.47351749621617x_{25} = -2.47351749621617
x25=−39.568478688083x_{25} = -39.568478688083
x25=−64.6547752099016x_{25} = -64.6547752099016
x25=−58.3805640516404x_{25} = -58.3805640516404
x25=55.2439678724582x_{25} = 55.2439678724582
x25=67.7923138841664x_{25} = 67.7923138841664
x25=−45.8368540585271x_{25} = -45.8368540585271
x25=−77.2063178228991x_{25} = -77.2063178228991
x25=42.7022928266836x_{25} = 42.7022928266836
Decreasing at intervals
[89.3099503974961,∞)\left[89.3099503974961, \infty\right)
Increasing at intervals
(−∞,−98.7422477693976]\left(-\infty, -98.7422477693976\right]
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
sin⁡(x)+2(tan⁡2(x)+1)tan⁡(x)x−2tan⁡2(x)x2−2(x−tan⁡(x))x3=0\sin{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{2 \tan^{2}{\left(x \right)}}{x^{2}} - \frac{2 \left(x - \tan{\left(x \right)}\right)}{x^{3}} = 0
Solve this equation
The roots of this equation
x1=−3.25531917889782x_{1} = -3.25531917889782
x2=65.9729718428982x_{2} = 65.9729718428982
x3=8.54407100749472x_{3} = 8.54407100749472
x4=−47.1247537226222x_{4} = -47.1247537226222
x5=−94.2475495662009x_{5} = -94.2475495662009
x6=−67.858791372479x_{6} = -67.858791372479
x7=−13.5900595381389x_{7} = -13.5900595381389
x8=−59.6908035413816x_{8} = -59.6908035413816
x9=−65.9738917028545x_{9} = -65.9738917028545
x10=−75.3978622848622x_{10} = -75.3978622848622
x11=−5.57723011480009x_{11} = -5.57723011480009
x12=56.5492718203601x_{12} = 56.5492718203601
x13=−72.2570037763859x_{13} = -72.2570037763859
x14=31.4178313425301x_{14} = 31.4178313425301
x15=72.2562370543402x_{15} = 72.2562370543402
x16=−87.9643298122186x_{16} = -87.9643298122186
x17=−17.7882924548015x_{17} = -17.7882924548015
x18=96.0975205871665x_{18} = 96.0975205871665
x19=−40.8418474931485x_{19} = -40.8418474931485
x20=−24.0171885604099x_{20} = -24.0171885604099
x21=100.531158946543x_{21} = 100.531158946543
x22=−84.8232732142152x_{22} = -84.8232732142152
x23=9.39587064204742x_{23} = 9.39587064204742
x24=−6.20464758748025x_{24} = -6.20464758748025
x25=−53.4077509603889x_{25} = -53.4077509603889
x26=−31.4137618644761x_{26} = -31.4137618644761
x27=84.8227169584775x_{27} = 84.8227169584775
x28=−37.6976255823892x_{28} = -37.6976255823892
x29=52.1810046823635x_{29} = 52.1810046823635
x30=−19.9416480661181x_{30} = -19.9416480661181
x31=−9.44325542478636x_{31} = -9.44325542478636
x32=59.6896796042739x_{32} = 59.6896796042739
x33=97.3891569725204x_{33} = 97.3891569725204
x34=−43.9812139241506x_{34} = -43.9812139241506
x35=28.2716408771708x_{35} = 28.2716408771708
x36=34.5557412938474x_{36} = 34.5557412938474
x37=−78.5401325124149x_{37} = -78.5401325124149
x38=−56.5480193708747x_{38} = -56.5480193708747
x39=69.1154452797118x_{39} = 69.1154452797118
x40=−97.389578882901x_{40} = -97.389578882901
x41=42.0419117735865x_{41} = 42.0419117735865
x42=62.8323440381529x_{42} = 62.8323440381529
x43=78.5394836355139x_{43} = 78.5394836355139
x44=−69.1146072115866x_{44} = -69.1146072115866
x45=−28.2766698147698x_{45} = -28.2766698147698
x46=58.4507796694138x_{46} = 58.4507796694138
x47=48.3424609874797x_{47} = 48.3424609874797
x48=75.3985664003351x_{48} = 75.3985664003351
x49=−21.9949375289679x_{49} = -21.9949375289679
x50=−12.5512452233507x_{50} = -12.5512452233507
x51=21.9865962327517x_{51} = 21.9865962327517
x52=53.406346615999x_{52} = 53.406346615999
x53=86.1047982542158x_{53} = 86.1047982542158
x54=87.9648470246859x_{54} = 87.9648470246859
x55=94.2480000856356x_{55} = 94.2480000856356
x56=81.6817015925809x_{56} = 81.6817015925809
x57=15.6986564968084x_{57} = 15.6986564968084
x58=3.89268236851131x_{58} = 3.89268236851131
x59=37.7004480494209x_{59} = 37.7004480494209
x60=−74.1324836563162x_{60} = -74.1324836563162
x61=14.6860597173348x_{61} = 14.6860597173348
x62=−30.2635250447958x_{62} = -30.2635250447958
x63=−34.5591020888815x_{63} = -34.5591020888815
x64=12.5772709635484x_{64} = 12.5772709635484
x65=−91.1064227304668x_{65} = -91.1064227304668
x66=−57.7881984974094x_{66} = -57.7881984974094
x67=150.796534173342x_{67} = 150.796534173342
x68=−100.530763003834x_{68} = -100.530763003834
x69=43.9832860061902x_{69} = 43.9832860061902
x70=40.8394435653021x_{70} = 40.8394435653021
x71=−25.1292996851753x_{71} = -25.1292996851753
x72=18.8546410666352x_{72} = 18.8546410666352
x73=25.1356731490023x_{73} = 25.1356731490023
x74=50.2662437051062x_{74} = 50.2662437051062
x75=6.3210262130017x_{75} = 6.3210262130017
x76=−81.6811016988525x_{76} = -81.6811016988525
x77=−15.7151443645601x_{77} = -15.7151443645601
x78=−50.2646580431382x_{78} = -50.2646580431382
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0x_{1} = 0

lim⁡x→0−(sin⁡(x)+2(tan⁡2(x)+1)tan⁡(x)x−2tan⁡2(x)x2−2(x−tan⁡(x))x3)=23\lim_{x \to 0^-}\left(\sin{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{2 \tan^{2}{\left(x \right)}}{x^{2}} - \frac{2 \left(x - \tan{\left(x \right)}\right)}{x^{3}}\right) = \frac{2}{3}
lim⁡x→0+(sin⁡(x)+2(tan⁡2(x)+1)tan⁡(x)x−2tan⁡2(x)x2−2(x−tan⁡(x))x3)=23\lim_{x \to 0^+}\left(\sin{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{2 \tan^{2}{\left(x \right)}}{x^{2}} - \frac{2 \left(x - \tan{\left(x \right)}\right)}{x^{3}}\right) = \frac{2}{3}
- limits are equal, then skip the corresponding point

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
[150.796534173342,∞)\left[150.796534173342, \infty\right)
Convex at the intervals
(−∞,−100.530763003834]\left(-\infty, -100.530763003834\right]
Vertical asymptotes
Have:
x1=0x_{1} = 0
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→−∞(−sin⁡(x)+−x+tan⁡(x)x)y = \lim_{x \to -\infty}\left(- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}\right)
True

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

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(−sin⁡(x)+−x+tan⁡(x)xx)y = x \lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x}}{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:
−sin⁡(x)+−x+tan⁡(x)x=sin⁡(x)−x−tan⁡(x)x- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x} = \sin{\left(x \right)} - \frac{x - \tan{\left(x \right)}}{x}
- No
−sin⁡(x)+−x+tan⁡(x)x=−sin⁡(x)+x−tan⁡(x)x- \sin{\left(x \right)} + \frac{- x + \tan{\left(x \right)}}{x} = - \sin{\left(x \right)} + \frac{x - \tan{\left(x \right)}}{x}
- No
so, the function
not is
neither even, nor odd
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