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  • How to use it?

  • Graphing y =:
  • x^3+4x
  • |x^2-x-2| |x^2-x-2|
  • x^2-6x+9
  • x^2+2x-4

  • Identical expressions

  • arctan(one /x)/(x+ two)
  • arc tangent of (1 divide by x) divide by (x plus 2)
  • arc tangent of (one divide by x) divide by (x plus two)
  • arctan1/x/x+2
  • arctan(1 divide by x) divide by (x+2)

  • Similar expressions

  • arctan(1/x)/(x-2)
Plotting a function graph/ arctan(1/x)/(x+2)

Graphing y = arctan(1/x)/(x+2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           /1\
       atan|-|
           \x/
f(x) = -------
        x + 2
f(x)=atan⁡(1x)x+2f{\left(x \right)} = \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + 2} 
f = atan(1/x)/(x + 2)
The graph of the function
02468-8-6-4-2-1010-2020
The domain of the function
The points at which the function is not precisely defined:
x1=−2x_{1} = -2
x2=0x_{2} = 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:
atan⁡(1x)x+2=0\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + 2} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to atan(1/x)/(x + 2).
atan⁡(10)2\frac{\operatorname{atan}{\left(\frac{1}{0} \right)}}{2}
The result:
f(0)=⟨−π4,π4⟩f{\left(0 \right)} = \left\langle - \frac{\pi}{4}, \frac{\pi}{4}\right\rangle
The point:
(0, AccumBounds(-pi/4, pi/4))
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
−atan⁡(1x)(x+2)2−1x2(1+1x2)(x+2)=0- \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} = 0
Solve this equation
The roots of this equation
x1=16587.7229190396x_{1} = 16587.7229190396
x2=36933.0242421313x_{2} = 36933.0242421313
x3=42866.5779293747x_{3} = 42866.5779293747
x4=25065.4688242842x_{4} = 25065.4688242842
x5=−18153.7968768281x_{5} = -18153.7968768281
x6=−35956.0390656664x_{6} = -35956.0390656664
x7=−19849.451180785x_{7} = -19849.451180785
x8=−38499.021037117x_{8} = -38499.021037117
x9=12348.0239520496x_{9} = 12348.0239520496
x10=30151.6672917348x_{10} = 30151.6672917348
x11=35237.7035205466x_{11} = 35237.7035205466
x12=29303.981040389x_{12} = 29303.981040389
x13=−41889.6312661903x_{13} = -41889.6312661903
x14=38628.3352801573x_{14} = 38628.3352801573
x15=28456.2900188718x_{15} = 28456.2900188718
x16=23370.015526887x_{16} = 23370.015526887
x17=−14762.1702224834x_{17} = -14762.1702224834
x18=−34260.7035621536x_{18} = -34260.7035621536
x19=−40194.3302854141x_{19} = -40194.3302854141
x20=−35108.372872x_{20} = -35108.372872
x21=−30022.3008645925x_{21} = -30022.3008645925
x22=36085.3651754688x_{22} = 36085.3651754688
x23=−25783.7700371527x_{23} = -25783.7700371527
x24=−13914.1680598466x_{24} = -13914.1680598466
x25=−37651.3629585359x_{25} = -37651.3629585359
x26=−17305.9359759458x_{26} = -17305.9359759458
x27=40323.6378353104x_{27} = 40323.6378353104
x28=27608.5937990085x_{28} = 27608.5937990085
x29=−23240.5602094796x_{29} = -23240.5602094796
x30=−24088.3063452608x_{30} = -24088.3063452608
x31=−27479.2016118189x_{31} = -27479.2016118189
x32=13196.0450486925x_{32} = 13196.0450486925
x33=−39346.6767624349x_{33} = -39346.6767624349
x34=42018.9329177414x_{34} = 42018.9329177414
x35=22522.2754805973x_{35} = 22522.2754805973
x36=−21545.0336007701x_{36} = -21545.0336007701
x37=24217.7463381975x_{37} = 24217.7463381975
x38=−12217.9913387618x_{38} = -12217.9913387618
x39=39475.9875509635x_{39} = 39475.9875509635
x40=21674.5251522753x_{40} = 21674.5251522753
x41=−29174.6067691777x_{41} = -29174.6067691777
x42=−36803.7023616607x_{42} = -36803.7023616607
x43=41171.2862538706x_{43} = 41171.2862538706
x44=17435.5690134626x_{44} = 17435.5690134626
x45=26760.8918998626x_{45} = 26760.8918998626
x46=−26631.4894482298x_{46} = -26631.4894482298
x47=18283.3935690576x_{47} = 18283.3935690576
x48=25913.1837793627x_{48} = 25913.1837793627
x49=30999.349155358x_{49} = 30999.349155358
x50=−24936.0426238831x_{50} = -24936.0426238831
x51=−22392.8030712172x_{51} = -22392.8030712172
x52=−13066.1122911295x_{52} = -13066.1122911295
x53=34390.0390897917x_{53} = 34390.0390897917
x54=19131.1993334596x_{54} = 19131.1993334596
x55=20826.7633307161x_{55} = 20826.7633307161
x56=−15610.1277726359x_{56} = -15610.1277726359
x57=−20697.2502454371x_{57} = -20697.2502454371
x58=31847.0269739212x_{58} = 31847.0269739212
x59=14044.018564154x_{59} = 14044.018564154
x60=−28326.9071906733x_{60} = -28326.9071906733
x61=−19001.634248567x_{61} = -19001.634248567
x62=14891.9521692111x_{62} = 14891.9521692111
x63=−33413.0308950943x_{63} = -33413.0308950943
x64=−41041.9817443474x_{64} = -41041.9817443474
x65=−30869.9899364173x_{65} = -30869.9899364173
x66=−32565.3546042869x_{66} = -32565.3546042869
x67=33542.3716770708x_{67} = 33542.3716770708
x68=−31717.6743942306x_{68} = -31717.6743942306
x69=−16458.0478017161x_{69} = -16458.0478017161
x70=19978.9886071672x_{70} = 19978.9886071672
x71=32694.7010553391x_{71} = 32694.7010553391
x72=15739.8519773568x_{72} = 15739.8519773568
x73=37780.6808915892x_{73} = 37780.6808915892
The values of the extrema at the points:
(16587.72291903956, 3.63390939414846e-9)

(36933.02424213132, 7.33072183377707e-10)

(42866.57792937472, 5.44179413311006e-10)

(25065.468824284155, 1.59152580333635e-9)

(-18153.796876828063, 3.03468001564172e-9)

(-35956.03906566644, 7.73535893033487e-10)

(-19849.451180785014, 2.53832223355915e-9)

(-38499.02103711696, 6.7471938717723e-10)

(12348.023952049642, 6.55744619684061e-9)

(30151.667291734844, 1.09988816496254e-9)

(35237.703520546616, 8.05304557132222e-10)

(29303.98104038901, 1.16443998730426e-9)

(-41889.63126619027, 5.69911815064452e-10)

(38628.33528015727, 6.70139982754219e-10)

(28456.290018871823, 1.23484634452392e-9)

(23370.015526887033, 1.83081656259178e-9)

(-14762.170222483426, 4.58942655745189e-9)

(-34260.70356215356, 8.51986667303278e-10)

(-40194.33028541407, 6.19001948967272e-10)

(-35108.37287200003, 8.1134083830826e-10)

(-30022.300864592482, 1.10953494899044e-9)

(36085.36517546877, 7.67916007881665e-10)

(-25783.770037152663, 1.50432215043283e-9)

(-13914.168059846603, 5.16592312791241e-9)

(-37651.36295853592, 7.05442573280625e-10)

(-17305.935975945784, 3.33933513746353e-9)

(40323.63783531036, 6.14977248444631e-10)

(27608.593799008457, 1.31183727205356e-9)

(-23240.560209479583, 1.85158728724227e-9)

(-24088.306345260837, 1.72354857755394e-9)

(-27479.201611818877, 1.3244128597311e-9)

(13196.045048692522, 5.74178072835781e-9)

(-39346.676762434894, 6.45960497977612e-10)

(42018.93291774137, 5.66355719448469e-10)

(22522.275480597287, 1.97122819879297e-9)

(-21545.033600770126, 2.15449735440318e-9)

(24217.7463381975, 1.70489125630029e-9)

(-12217.991338761805, 6.69994858512238e-9)

(39475.98755096351, 6.41670377669519e-10)

(21674.525152275255, 2.12843672576819e-9)

(-29174.606769177743, 1.17495098782929e-9)

(-36803.70236166067, 7.38313112289218e-10)

(41171.28625387064, 5.89915806463741e-10)

(17435.569013462577, 3.28910649123316e-9)

(26760.891899862592, 1.39626025604068e-9)

(-26631.48944822983, 1.41007334595823e-9)

(18283.393569057633, 2.99115472903318e-9)

(25913.183779362742, 1.48910366201143e-9)

(30999.349155358046, 1.04055928711977e-9)

(-24936.042623883113, 1.60834706395684e-9)

(-22392.803071217215, 1.99444410054978e-9)

(-13066.112291129475, 5.85832833253184e-9)

(34390.03908979168, 8.45491809216041e-10)

(19131.199333459597, 2.73193402262787e-9)

(20826.763330716087, 2.30523249325733e-9)

(-15610.127772635873, 4.10433432669865e-9)

(-20697.250245437055, 2.33462250029408e-9)

(31847.026973921213, 9.85904696571643e-10)

(14044.01856415396, 5.06938606079722e-9)

(-28326.907190673326, 1.24632797891631e-9)

(-19001.634248567043, 2.76989817779042e-9)

(14891.952169211147, 4.50856579523168e-9)

(-33413.03089509429, 8.95765334698308e-10)

(-41041.981744347395, 5.93696540305781e-10)

(-30869.989936417267, 1.0494340759053e-9)

(-32565.35460428687, 9.43007289173052e-10)

(33542.37167707083, 8.88764238480341e-10)

(-31717.674394230628, 9.94087728879106e-10)

(-16458.04780171614, 3.69229285477188e-9)

(19978.988607167194, 2.50501037096808e-9)

(32694.7010553391, 9.35445935974442e-10)

(15739.851977356784, 4.03592912764465e-9)

(37780.68089158923, 7.00547276479032e-10)


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:
The function has no minima
The function has no maxima
Decreasing at the entire real axis
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
2(atan⁡(1x)(x+2)2+1x2(1+1x2)(x+2)+1−1x2(1+1x2)x3(1+1x2))x+2=0\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2} = 0
Solve this equation
The roots of this equation
x1=4266.65367445192x_{1} = 4266.65367445192
x2=9937.37395136824x_{2} = 9937.37395136824
x3=3612.17008848393x_{3} = 3612.17008848393
x4=9719.28819648547x_{4} = 9719.28819648547
x5=−4452.88432742144x_{5} = -4452.88432742144
x6=−10559.8323524182x_{6} = -10559.8323524182
x7=10155.4590528456x_{7} = 10155.4590528456
x8=5139.19521736223x_{8} = 5139.19521736223
x9=−4671.03531898808x_{9} = -4671.03531898808
x10=−1833.45731857528x_{10} = -1833.45731857528
x11=7320.28554933361x_{11} = 7320.28554933361
x12=−4889.17887858538x_{12} = -4889.17887858538
x13=−3143.73181577214x_{13} = -3143.73181577214
x14=5793.55185085321x_{14} = 5793.55185085321
x15=−2488.85074573824x_{15} = -2488.85074573824
x16=−6197.92913031119x_{16} = -6197.92913031119
x17=−8597.0397249602x_{17} = -8597.0397249602
x18=8192.66534690497x_{18} = 8192.66534690497
x19=7974.57237255382x_{19} = 7974.57237255382
x20=6229.77317138162x_{20} = 6229.77317138162
x21=−9905.57533377917x_{21} = -9905.57533377917
x22=−5325.44745340199x_{22} = -5325.44745340199
x23=10809.710824617x_{23} = 10809.710824617
x24=−5543.57396632486x_{24} = -5543.57396632486
x25=−4234.72471680378x_{25} = -4234.72471680378
x26=11027.7936852099x_{26} = 11027.7936852099
x27=−5107.31598521205x_{27} = -5107.31598521205
x28=2084.45236245165x_{28} = 2084.45236245165
x29=2957.5753803023x_{29} = 2957.5753803023
x30=2739.34003587x_{30} = 2739.34003587
x31=1866.06649845331x_{31} = 1866.06649845331
x32=−7942.75750807917x_{32} = -7942.75750807917
x33=−8378.94691851695x_{33} = -8378.94691851695
x34=−9687.48824590579x_{34} = -9687.48824590579
x35=7538.38258474053x_{35} = 7538.38258474053
x36=9283.11454648586x_{36} = 9283.11454648586
x37=8410.75717085281x_{37} = 8410.75717085281
x38=4484.79808915367x_{38} = 4484.79808915367
x39=−5761.69610121046x_{39} = -5761.69610121046
x40=−10123.6616831784x_{40} = -10123.6616831784
x41=−2707.18224487169x_{41} = -2707.18224487169
x42=−9469.4003676517x_{42} = -9469.4003676517
x43=−4016.55503372661x_{43} = -4016.55503372661
x44=3830.34101271359x_{44} = 3830.34101271359
x45=6665.98425095459x_{45} = 6665.98425095459
x46=7102.18691631106x_{46} = 7102.18691631106
x47=3175.78984726872x_{47} = 3175.78984726872
x48=−9251.31164213201x_{48} = -9251.31164213201
x49=10591.6274538209x_{49} = 10591.6274538209
x50=10373.5435412909x_{50} = 10373.5435412909
x51=2302.78597024282x_{51} = 2302.78597024282
x52=5357.3178943352x_{52} = 5357.3178943352
x53=4048.50171707908x_{53} = 4048.50171707908
x54=6884.08653869763x_{54} = 6884.08653869763
x55=−2052.01060774317x_{55} = -2052.01060774317
x56=9065.02655079485x_{56} = 9065.02655079485
x57=−10341.7473415644x_{57} = -10341.7473415644
x58=−7288.46215665169x_{58} = -7288.46215665169
x59=−5979.81434909453x_{59} = -5979.81434909453
x60=−6852.25605773224x_{60} = -6852.25605773224
x61=−7070.36014300167x_{61} = -7070.36014300167
x62=−2925.47299918569x_{62} = -2925.47299918569
x63=−8815.13139310588x_{63} = -8815.13139310588
x64=−7724.66068520671x_{64} = -7724.66068520671
x65=4921.06809983418x_{65} = 4921.06809983418
x66=−7506.56228301616x_{66} = -7506.56228301616
x67=7756.47815347094x_{67} = 7756.47815347094
x68=−3580.17780168701x_{68} = -3580.17780168701
x69=−10777.9167556495x_{69} = -10777.9167556495
x70=−3798.37347953912x_{70} = -3798.37347953912
x71=3393.98712459383x_{71} = 3393.98712459383
x72=8846.93769953506x_{72} = 8846.93769953506
x73=6447.87986596005x_{73} = 6447.87986596005
x74=−6416.04080676295x_{74} = -6416.04080676295
x75=−1614.76423318349x_{75} = -1614.76423318349
x76=2521.07901482993x_{76} = 2521.07901482993
x77=5575.43662977994x_{77} = 5575.43662977994
x78=−8160.85288071644x_{78} = -8160.85288071644
x79=9501.20174420201x_{79} = 9501.20174420201
x80=4702.93595515051x_{80} = 4702.93595515051
x81=1647.6121150313x_{81} = 1647.6121150313
x82=−9033.22200687244x_{82} = -9033.22200687244
x83=−2270.4661524954x_{83} = -2270.4661524954
x84=−3361.96514138765x_{84} = -3361.96514138765
x85=−6634.14969172341x_{85} = -6634.14969172341
x86=8628.84792943952x_{86} = 8628.84792943952
x87=6011.6639252941x_{87} = 6011.6639252941
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=−2x_{1} = -2
x2=0x_{2} = 0

lim⁡x→−2−(2(atan⁡(1x)(x+2)2+1x2(1+1x2)(x+2)+1−1x2(1+1x2)x3(1+1x2))x+2)=∞\lim_{x \to -2^-}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = \infty
lim⁡x→−2+(2(atan⁡(1x)(x+2)2+1x2(1+1x2)(x+2)+1−1x2(1+1x2)x3(1+1x2))x+2)=−∞\lim_{x \to -2^+}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = -\infty
- the limits are not equal, so
x1=−2x_{1} = -2
- is an inflection point
lim⁡x→0−(2(atan⁡(1x)(x+2)2+1x2(1+1x2)(x+2)+1−1x2(1+1x2)x3(1+1x2))x+2)=12−π8\lim_{x \to 0^-}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = \frac{1}{2} - \frac{\pi}{8}
lim⁡x→0+(2(atan⁡(1x)(x+2)2+1x2(1+1x2)(x+2)+1−1x2(1+1x2)x3(1+1x2))x+2)=π8+12\lim_{x \to 0^+}\left(\frac{2 \left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + 2\right)^{2}} + \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + 2\right)} + \frac{1 - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}}{x^{3} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x + 2}\right) = \frac{\pi}{8} + \frac{1}{2}
- the limits are not equal, so
x2=0x_{2} = 0
- is an inflection 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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
x1=−2x_{1} = -2
x2=0x_{2} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(atan⁡(1x)x+2)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + 2}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(atan⁡(1x)x+2)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + 2}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(1/x)/(x + 2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(atan⁡(1x)x(x+2))=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x \left(x + 2\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(atan⁡(1x)x(x+2))=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x \left(x + 2\right)}\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:
atan⁡(1x)x+2=−atan⁡(1x)2−x\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + 2} = - \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{2 - x}
- No
atan⁡(1x)x+2=atan⁡(1x)2−x\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + 2} = \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{2 - x}
- No
so, the function
not is
neither even, nor odd
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