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Plotting a function graph/ (arctg(x-1)/(x^2-1))

Graphing y = (arctg(x-1)/(x^2-1))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       atan(x - 1)
f(x) = -----------
           2      
          x  - 1
f(x)=atan⁡(x−1)x2−1f{\left(x \right)} = \frac{\operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1} 
f = atan(x - 1)/(x^2 - 1)
The graph of the function
02468-8-6-4-2-1010-2525
The domain of the function
The points at which the function is not precisely defined:
x1=−1x_{1} = -1
x2=1x_{2} = 1
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⁡(x−1)x2−1=0\frac{\operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1} = 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(x - 1)/(x^2 - 1).
atan⁡(−1)−1+02\frac{\operatorname{atan}{\left(-1 \right)}}{-1 + 0^{2}}
The result:
f(0)=π4f{\left(0 \right)} = \frac{\pi}{4}
The point:
(0, 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
−2xatan⁡(x−1)(x2−1)2+1(x2−1)((x−1)2+1)=0- \frac{2 x \operatorname{atan}{\left(x - 1 \right)}}{\left(x^{2} - 1\right)^{2}} + \frac{1}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} = 0
Solve this equation
The roots of this equation
x1=41732.6951021162x_{1} = 41732.6951021162
x2=40037.4837685001x_{2} = 40037.4837685001
x3=27323.3984162899x_{3} = 27323.3984162899
x4=−28881.1962419616x_{4} = -28881.1962419616
x5=20542.5636397183x_{5} = 20542.5636397183
x6=−15313.819694531x_{6} = -15313.819694531
x7=28171.0038462628x_{7} = 28171.0038462628
x8=38342.2723552181x_{8} = 38342.2723552181
x9=21390.1665034433x_{9} = 21390.1665034433
x10=−42444.8970797256x_{10} = -42444.8970797256
x11=23085.3739636572x_{11} = 23085.3739636572
x12=31561.4263435655x_{12} = 31561.4263435655
x13=13761.7961969892x_{13} = 13761.7961969892
x14=−23794.2027929786x_{14} = -23794.2027929786
x15=−30576.7585274643x_{15} = -30576.7585274643
x16=34951.8493563508x_{16} = 34951.8493563508
x17=−38206.3969446393x_{17} = -38206.3969446393
x18=−28033.3988668025x_{18} = -28033.3988668025
x19=−34815.5120862654x_{19} = -34815.5120862654
x20=26475.7931123953x_{20} = 26475.7931123953
x21=−13617.0799980095x_{21} = -13617.0799980095
x22=−37358.6837623526x_{22} = -37358.6837623526
x23=24780.5830260884x_{23} = 24780.5830260884
x24=42580.3007360941x_{24} = 42580.3007360941
x25=35799.4551200724x_{25} = 35799.4551200724
x26=−18706.4528567436x_{26} = -18706.4528567436
x27=−17858.3748250259x_{27} = -17858.3748250259
x28=39189.8780710616x_{28} = 39189.8780710616
x29=30713.8206402264x_{29} = 30713.8206402264
x30=37494.66662299x_{30} = 37494.66662299
x31=−27185.5893046584x_{31} = -27185.5893046584
x32=−36510.9654760296x_{32} = -36510.9654760296
x33=−29728.9824836733x_{33} = -29728.9824836733
x34=29866.2149788368x_{34} = 29866.2149788368
x35=−21250.4568264737x_{35} = -21250.4568264737
x36=18847.3604443445x_{36} = 18847.3604443445
x37=−39054.1053579782x_{37} = -39054.1053579782
x38=−41597.2049223586x_{38} = -41597.2049223586
x39=−33967.776135552x_{39} = -33967.776135552
x40=−26337.7663647711x_{40} = -26337.7663647711
x41=−40749.5090776704x_{41} = -40749.5090776704
x42=32409.0320762992x_{42} = 32409.0320762992
x43=−32272.2832672024x_{43} = -32272.2832672024
x44=−25489.928696076x_{44} = -25489.928696076
x45=−19554.4891574774x_{45} = -19554.4891574774
x46=34104.2435904426x_{46} = 34104.2435904426
x47=−22946.3107719604x_{47} = -22946.3107719604
x48=17999.7605732465x_{48} = 17999.7605732465
x49=−17010.2487134799x_{49} = -17010.2487134799
x50=−35663.2417184403x_{50} = -35663.2417184403
x51=−24642.0747592927x_{51} = -24642.0747592927
x52=15456.9723148165x_{52} = 15456.9723148165
x53=17152.1622761647x_{53} = 17152.1622761647
x54=33256.6378280935x_{54} = 33256.6378280935
x55=16304.5659946845x_{55} = 16304.5659946845
x56=40885.0894459342x_{56} = 40885.0894459342
x57=23932.9783366318x_{57} = 23932.9783366318
x58=29018.6093746581x_{58} = 29018.6093746581
x59=−22098.3963560749x_{59} = -22098.3963560749
x60=14609.3820242319x_{60} = 14609.3820242319
x61=−33120.0333764647x_{61} = -33120.0333764647
x62=−20402.4890058282x_{62} = -20402.4890058282
x63=−39901.8093087569x_{63} = -39901.8093087569
x64=−14465.4956454721x_{64} = -14465.4956454721
x65=36647.0608769001x_{65} = 36647.0608769001
x66=25628.1879684537x_{66} = 25628.1879684537
x67=19694.9615528886x_{67} = 19694.9615528886
x68=−31424.5252069205x_{68} = -31424.5252069205
x69=−16162.0668174733x_{69} = -16162.0668174733
x70=22237.7699862725x_{70} = 22237.7699862725
The values of the extrema at the points:
(41732.695102116224, 9.01904154045824e-10)

(40037.48376850013, 9.79894726253857e-10)

(27323.39841628989, 2.10397380594579e-9)

(-28881.196241961603, -1.88312847211983e-9)

(20542.563639718326, 3.72217798511809e-9)

(-15313.819694530977, -6.69783996646637e-9)

(28171.00384626281, 1.97927170833934e-9)

(38342.27235521808, 1.06845680807748e-9)

(21390.16650344333, 3.43303808389221e-9)

(-42444.89707972562, -8.71891399699013e-10)

(23085.373963657177, 2.94736602114959e-9)

(31561.42634356547, 1.57687719810736e-9)

(13761.796196989177, 8.29372331882375e-9)

(-23794.20279297856, -2.77437997498929e-9)

(-30576.75852746425, -1.68007221090125e-9)

(34951.84935635075, 1.2857947465329e-9)

(-38206.396944639346, -1.07606987675392e-9)

(-28033.39886680253, -1.99875011909678e-9)

(-34815.512086265415, -1.29588469724359e-9)

(26475.79311239531, 2.24084324347717e-9)

(-13617.079998009467, -8.47093986472269e-9)

(-37358.68376235261, -1.1254581323037e-9)

(24780.583026088356, 2.55791247461768e-9)

(42580.30073609408, 8.66355097417115e-10)

(35799.45512007243, 1.22562982544271e-9)

(-18706.452856743646, -4.48871685490992e-9)

(-17858.37482502587, -4.92516210292849e-9)

(39189.878071061554, 1.02273942491395e-9)

(30713.820640226408, 1.66511099179531e-9)

(37494.66662298999, 1.11730954840083e-9)

(-27185.589304658384, -2.12535853990813e-9)

(-36510.96547602957, -1.17832656892473e-9)

(-29728.982483673324, -1.77725803955371e-9)

(29866.214978836815, 1.76096306125911e-9)

(-21250.456826473743, -3.47832634864163e-9)

(18847.360444344504, 4.42185131453689e-9)

(-39054.10535797821, -1.02986289436115e-9)

(-41597.204922358644, -9.07789032375997e-10)

(-33967.77613555197, -1.36137417807154e-9)

(-26337.766364771138, -2.26439141024015e-9)

(-40749.5090776704, -9.45950324408325e-10)

(32409.03207629917, 1.49547523260251e-9)

(-32272.283267202383, -1.50817565032972e-9)

(-25489.92869607596, -2.41752954367036e-9)

(-19554.48915747735, -4.10783309851236e-9)

(34104.243590442595, 1.35050105238038e-9)

(-22946.310771960383, -2.98319805320524e-9)

(17999.760573246498, 4.84809431934978e-9)

(-17010.248713479854, -5.42853069862973e-9)

(-35663.24171844028, -1.23501004708466e-9)

(-24642.07475929272, -2.58674796794428e-9)

(15456.972314816516, 6.57435470979148e-9)

(17152.162276164687, 5.33907478644989e-9)

(33256.63782809345, 1.42021757196306e-9)

(16304.565994684515, 5.90859787730375e-9)

(40885.08944593416, 9.39686981765065e-10)

(23932.97833663176, 2.74229915265772e-9)

(29018.609374658146, 1.86533636353917e-9)

(-22098.39635607487, -3.21651698731235e-9)

(14609.382024231918, 7.35931277536362e-9)

(-33120.03337646466, -1.43195707061265e-9)

(-20402.48900582819, -3.77346235927944e-9)

(-39901.80930875689, -9.86569694867799e-10)

(-14465.49564547205, -7.50644192786878e-9)

(36647.06087690013, 1.16959104231041e-9)

(25628.187968453716, 2.39151598311515e-9)

(19694.961552888588, 4.04944555644357e-9)

(-31424.52520692052, -1.5906463385201e-9)

(-16162.066817473331, -6.01324608034016e-9)

(22237.769986272524, 3.1763253198712e-9)


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(−2x(x2−1)((x−1)2+1)−x−1((x−1)2+1)2+(4x2x2−1−1)atan⁡(x−1)x2−1)x2−1=0\frac{2 \left(- \frac{2 x}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} - \frac{x - 1}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right)}{x^{2} - 1} = 0
Solve this equation
The roots of this equation
x1=6811.41815542743x_{1} = 6811.41815542743
x2=2668.19670006129x_{2} = 2668.19670006129
x3=−2847.16269001367x_{3} = -2847.16269001367
x4=10518.614624977x_{4} = 10518.614624977
x5=−5902.90626432681x_{5} = -5902.90626432681
x6=8119.83948181682x_{6} = 8119.83948181682
x7=4630.72814400825x_{7} = 4630.72814400825
x8=3540.40648032681x_{8} = 3540.40648032681
x9=5721.06977649485x_{9} = 5721.06977649485
x10=9864.40309784326x_{10} = 9864.40309784326
x11=−4157.27399859556x_{11} = -4157.27399859556
x12=−7648.04425945574x_{12} = -7648.04425945574
x13=−9392.96605534623x_{13} = -9392.96605534623
x14=−9829.1763422554x_{14} = -9829.1763422554
x15=−2628.64211378136x_{15} = -2628.64211378136
x16=9428.26210891638x_{16} = 9428.26210891638
x17=7465.62855838261x_{17} = 7465.62855838261
x18=−1972.45987214506x_{18} = -1972.45987214506
x19=−3502.38001257541x_{19} = -3502.38001257541
x20=7901.76913183029x_{20} = 7901.76913183029
x21=−8738.63741065811x_{21} = -8738.63741065811
x22=2232.13893691943x_{22} = 2232.13893691943
x23=−10483.4807773401x_{23} = -10483.4807773401
x24=−3720.70532998241x_{24} = -3720.70532998241
x25=−6121.06675052115x_{25} = -6121.06675052115
x26=4848.79553643877x_{26} = 4848.79553643877
x27=−4811.9787708472x_{27} = -4811.9787708472
x28=−7211.78623703032x_{28} = -7211.78623703032
x29=10082.4736025948x_{29} = 10082.4736025948
x30=−7429.9169956772x_{30} = -7429.9169956772
x31=−10047.2791902854x_{31} = -10047.2791902854
x32=4194.59566306667x_{32} = 4194.59566306667
x33=−9174.85839596281x_{33} = -9174.85839596281
x34=7029.48821321239x_{34} = 7029.48821321239
x35=−2191.32254373413x_{35} = -2191.32254373413
x36=−8956.74889420891x_{36} = -8956.74889420891
x37=5066.86349433144x_{37} = 5066.86349433144
x38=−3939.00178115969x_{38} = -3939.00178115969
x39=−3284.01993747129x_{39} = -3284.01993747129
x40=−6339.22093528361x_{40} = -6339.22093528361
x41=−10701.5796853305x_{41} = -10701.5796853305
x42=−4375.52567475457x_{42} = -4375.52567475457
x43=7247.55835195549x_{43} = 7247.55835195549
x44=−8520.52379145514x_{44} = -8520.52379145514
x45=6593.34819458712x_{45} = 6593.34819458712
x46=3976.53100048177x_{46} = 3976.53100048177
x47=5503.00069140277x_{47} = 5503.00069140277
x48=−1753.39222987364x_{48} = -1753.39222987364
x49=3322.34766346881x_{49} = 3322.34766346881
x50=−5466.56332152108x_{50} = -5466.56332152108
x51=−6775.51286689925x_{51} = -6775.51286689925
x52=7683.69882142747x_{52} = 7683.69882142747
x53=10954.7556567109x_{53} = 10954.7556567109
x54=−6557.36945599108x_{54} = -6557.36945599108
x55=−7866.16832246646x_{55} = -7866.16832246646
x56=−9611.07199900913x_{56} = -9611.07199900913
x57=−6993.65165225979x_{57} = -6993.65165225979
x58=−10919.6774375952x_{58} = -10919.6774375952
x59=4412.66145827765x_{59} = 4412.66145827765
x60=10736.6851401751x_{60} = 10736.6851401751
x61=8555.98027537042x_{61} = 8555.98027537042
x62=8774.05070872731x_{62} = 8774.05070872731
x63=1796.16694984546x_{63} = 1796.16694984546
x64=6375.27835002919x_{64} = 6375.27835002919
x65=3104.29218958228x_{65} = 3104.29218958228
x66=5284.93190799427x_{66} = 5284.93190799427
x67=2014.13690397986x_{67} = 2014.13690397986
x68=−2410.03737475158x_{68} = -2410.03737475158
x69=9646.3325993535x_{69} = 9646.3325993535
x70=9210.19162862696x_{70} = 9210.19162862696
x71=3758.46779365714x_{71} = 3758.46779365714
x72=−3065.61749619387x_{72} = -3065.61749619387
x73=8992.12116093852x_{73} = 8992.12116093852
x74=−4593.75978881552x_{74} = -4593.75978881552
x75=−5030.18462266333x_{75} = -5030.18462266333
x76=8337.90986483989x_{76} = 8337.90986483989
x77=−8302.40786640037x_{77} = -8302.40786640037
x78=10300.5441120764x_{78} = 10300.5441120764
x79=6157.20864525689x_{79} = 6157.20864525689
x80=−5684.73874002286x_{80} = -5684.73874002286
x81=0.679997123576428x_{81} = 0.679997123576428
x82=2450.16117528235x_{82} = 2450.16117528235
x83=5939.1391089851x_{83} = 5939.1391089851
x84=−5248.37900812319x_{84} = -5248.37900812319
x85=−8084.28944672083x_{85} = -8084.28944672083
x86=2886.24127305287x_{86} = 2886.24127305287
x87=−10265.3806392786x_{87} = -10265.3806392786
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=−1x_{1} = -1
x2=1x_{2} = 1

lim⁡x→−1−(2(−2x(x2−1)((x−1)2+1)−x−1((x−1)2+1)2+(4x2x2−1−1)atan⁡(x−1)x2−1)x2−1)=−∞\lim_{x \to -1^-}\left(\frac{2 \left(- \frac{2 x}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} - \frac{x - 1}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right)}{x^{2} - 1}\right) = -\infty
lim⁡x→−1+(2(−2x(x2−1)((x−1)2+1)−x−1((x−1)2+1)2+(4x2x2−1−1)atan⁡(x−1)x2−1)x2−1)=∞\lim_{x \to -1^+}\left(\frac{2 \left(- \frac{2 x}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} - \frac{x - 1}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right)}{x^{2} - 1}\right) = \infty
- the limits are not equal, so
x1=−1x_{1} = -1
- is an inflection point
lim⁡x→1−(2(−2x(x2−1)((x−1)2+1)−x−1((x−1)2+1)2+(4x2x2−1−1)atan⁡(x−1)x2−1)x2−1)=−0.0833333333333333\lim_{x \to 1^-}\left(\frac{2 \left(- \frac{2 x}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} - \frac{x - 1}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right)}{x^{2} - 1}\right) = -0.0833333333333333
lim⁡x→1+(2(−2x(x2−1)((x−1)2+1)−x−1((x−1)2+1)2+(4x2x2−1−1)atan⁡(x−1)x2−1)x2−1)=−0.0833333333333333\lim_{x \to 1^+}\left(\frac{2 \left(- \frac{2 x}{\left(x^{2} - 1\right) \left(\left(x - 1\right)^{2} + 1\right)} - \frac{x - 1}{\left(\left(x - 1\right)^{2} + 1\right)^{2}} + \frac{\left(\frac{4 x^{2}}{x^{2} - 1} - 1\right) \operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right)}{x^{2} - 1}\right) = -0.0833333333333333
- the limits are not equal, so
x2=1x_{2} = 1
- 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:
Concave at the intervals
(−∞,0.679997123576428]\left(-\infty, 0.679997123576428\right]
Convex at the intervals
[0.679997123576428,∞)\left[0.679997123576428, \infty\right)
Vertical asymptotes
Have:
x1=−1x_{1} = -1
x2=1x_{2} = 1
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(atan⁡(x−1)x2−1)=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(atan⁡(x−1)x2−1)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1}\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(x - 1)/(x^2 - 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(atan⁡(x−1)x(x2−1))=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(x - 1 \right)}}{x \left(x^{2} - 1\right)}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(atan⁡(x−1)x(x2−1))=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(x - 1 \right)}}{x \left(x^{2} - 1\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⁡(x−1)x2−1=−atan⁡(x+1)x2−1\frac{\operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1} = - \frac{\operatorname{atan}{\left(x + 1 \right)}}{x^{2} - 1}
- No
atan⁡(x−1)x2−1=atan⁡(x+1)x2−1\frac{\operatorname{atan}{\left(x - 1 \right)}}{x^{2} - 1} = \frac{\operatorname{atan}{\left(x + 1 \right)}}{x^{2} - 1}
- No
so, the function
not is
neither even, nor odd
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