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Plotting a function graph/ (x+1)arctg(x+1)

Graphing y = (x+1)arctg(x+1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = (x + 1)*atan(x + 1)
f(x)=(x+1)atan⁡(x+1)f{\left(x \right)} = \left(x + 1\right) \operatorname{atan}{\left(x + 1 \right)} 
f = (x + 1)*atan(x + 1)
The graph of the function
02468-8-6-4-2-1010020
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:
(x+1)atan⁡(x+1)=0\left(x + 1\right) \operatorname{atan}{\left(x + 1 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−1x_{1} = -1
Numerical solution
x1=−0.999999483228915x_{1} = -0.999999483228915
x2=−0.999999484804135x_{2} = -0.999999484804135
x3=−0.999999486509974x_{3} = -0.999999486509974
x4=−0.999999483628689x_{4} = -0.999999483628689
x5=−1.00000051686884x_{5} = -1.00000051686884
x6=−1.00000051628437x_{6} = -1.00000051628437
x7=−1.00000030086633x_{7} = -1.00000030086633
x8=−1.00000051708657x_{8} = -1.00000051708657
x9=−0.999999483259633x_{9} = -0.999999483259633
x10=−0.999999484087395x_{10} = -0.999999484087395
x11=−1.0000005167933x_{11} = -1.0000005167933
x12=−0.999999487357113x_{12} = -0.999999487357113
x13=−1.00000051514264x_{13} = -1.00000051514264
x14=−0.999999483200405x_{14} = -0.999999483200405
x15=−0.999999483867896x_{15} = -0.999999483867896
x16=−1.00000051653166x_{16} = -1.00000051653166
x17=−0.999999483083971x_{17} = -0.999999483083971
x18=−0.999999482933008x_{18} = -0.999999482933008
x19=−1.00000051694914x_{19} = -1.00000051694914
x20=−0.999999494031996x_{20} = -0.999999494031996
x21=−1.00000051642109x_{21} = -1.00000051642109
x22=−0.999999482998444x_{22} = -0.999999482998444
x23=−1.00000050507963x_{23} = -1.00000050507963
x24=−1.00000051703978x_{24} = -1.00000051703978
x25=−1.00000051574465x_{25} = -1.00000051574465
x26=−1.00000051709698x_{26} = -1.00000051709698
x27=−0.999999482957244x_{27} = -0.999999482957244
x28=−0.99999948290065x_{28} = -0.99999948290065
x29=−0.999999483778329x_{29} = -0.999999483778329
x30=−0.999999483969997x_{30} = -0.999999483969997
x31=−1.00000051588532x_{31} = -1.00000051588532
x32=−1.00000051706428x_{32} = -1.00000051706428
x33=−1.00000051666271x_{33} = -1.00000051666271
x34=−1.00000051647916x_{34} = -1.00000051647916
x35=−1.00000051710693x_{35} = -1.00000051710693
x36=−1.00000051538172x_{36} = -1.00000051538172
x37=−0.999999482944834x_{37} = -0.999999482944834
x38=−1.00000051611122x_{38} = -1.00000051611122
x39=−0.999999483029772x_{39} = -0.999999483029772
x40=−0.999999483292825x_{40} = -0.999999483292825
x41=−0.999999483173877x_{41} = -0.999999483173877
x42=−0.999999482881351x_{42} = -0.999999482881351
x43=−1.00000051698267x_{43} = -1.00000051698267
x44=−0.999999483410592x_{44} = -0.999999483410592
x45=−1.00000050895384x_{45} = -1.00000050895384
x46=−1.00000051110355x_{46} = -1.00000051110355
x47=−1.00000051682024x_{47} = -1.00000051682024
x48=−1.00000051705233x_{48} = -1.00000051705233
x49=−1.0000005169981x_{49} = -1.0000005169981
x50=−1.00000051689083x_{50} = -1.00000051689083
x51=−0.999999488599745x_{51} = -0.999999488599745
x52=−0.999999483367912x_{52} = -0.999999483367912
x53=−0.999999483046795x_{53} = -0.999999483046795
x54=−0.999999485442001x_{54} = -0.999999485442001
x55=−0.999999482970283x_{55} = -0.999999482970283
x56=−0.99999948589994x_{56} = -0.99999948589994
x57=−1.00000051447572x_{57} = -1.00000051447572
x58=−0.999999485086859x_{58} = -0.999999485086859
x59=−0.999999484383777x_{59} = -0.999999484383777
x60=−1.00000051600624x_{60} = -1.00000051600624
x61=−1.00000051673309x_{61} = -1.00000051673309
x62=−0.99999948422371x_{62} = -0.99999948422371
x63=−0.999999482983998x_{63} = -0.999999482983998
x64=−0.999999490564929x_{64} = -0.999999490564929
x65=−1.0000005148477x_{65} = -1.0000005148477
x66=−0.999999483457342x_{66} = -0.999999483457342
x67=−1.00000051676433x_{67} = -1.00000051676433
x68=−0.999999522095937x_{68} = -0.999999522095937
x69=−0.999999484574183x_{69} = -0.999999484574183
x70=−1.00000051669933x_{70} = -1.00000051669933
x71=−1.00000051693086x_{71} = -1.00000051693086
x72=−1.00000051657935x_{72} = -1.00000051657935
x73=−0.999999483699156x_{73} = -0.999999483699156
x74=−1.00000051635653x_{74} = -1.00000051635653
x75=−0.999999482890793x_{75} = -0.999999482890793
x76=−1.00000051334767x_{76} = -1.00000051334767
x77=−1.00000051620319x_{77} = -1.00000051620319
x78=−0.999999501301654x_{78} = -0.999999501301654
x79=−1.00000051702659x_{79} = -1.00000051702659
x80=−0.999999483125996x_{80} = -0.999999483125996
x81=−0.99999948310432x_{81} = -0.99999948310432
x82=−1.00000051557912x_{82} = -1.00000051557912
x83=−1.00000051696638x_{83} = -1.00000051696638
x84=−0.999999483565587x_{84} = -0.999999483565587
x85=−0.99999948306483x_{85} = -0.99999948306483
x86=−0.999999482921725x_{86} = -0.999999482921725
x87=−1.0000005139937x_{87} = -1.0000005139937
x88=−0.99999948301368x_{88} = -0.99999948301368
x89=−1.00000047089392x_{89} = -1.00000047089392
x90=−0.999999483149131x_{90} = -0.999999483149131
x91=−0.999999639752099x_{91} = -0.999999639752099
x92=−0.999999482910951x_{92} = -0.999999482910951
x93=−1.00000051691146x_{93} = -1.00000051691146
x94=−1.0000004966661x_{94} = -1.0000004966661
x95=−1.00000051244353x_{95} = -1.00000051244353
x96=−0.999999483328798x_{96} = -0.999999483328798
x97=−0.999999483508765x_{97} = -0.999999483508765
x98=−1.00000051707568x_{98} = -1.00000051707568
x99=−1.00000051701271x_{99} = -1.00000051701271
x100=−1.00000051684537x_{100} = -1.00000051684537
x101=−1.00000051662286x_{101} = -1.00000051662286
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x + 1)*atan(x + 1).
atan⁡(1)\operatorname{atan}{\left(1 \right)}
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
x+1(x+1)2+1+atan⁡(x+1)=0\frac{x + 1}{\left(x + 1\right)^{2} + 1} + \operatorname{atan}{\left(x + 1 \right)} = 0
Solve this equation
The roots of this equation
x1=−1x_{1} = -1
The values of the extrema at the points:
(-1, 0)


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=−1x_{1} = -1
The function has no maxima
Decreasing at intervals
[−1,∞)\left[-1, \infty\right)
Increasing at intervals
(−∞,−1]\left(-\infty, -1\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
2(−(x+1)2(x+1)2+1+1)(x+1)2+1=0\frac{2 \left(- \frac{\left(x + 1\right)^{2}}{\left(x + 1\right)^{2} + 1} + 1\right)}{\left(x + 1\right)^{2} + 1} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((x+1)atan⁡(x+1))=∞\lim_{x \to -\infty}\left(\left(x + 1\right) \operatorname{atan}{\left(x + 1 \right)}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞((x+1)atan⁡(x+1))=∞\lim_{x \to \infty}\left(\left(x + 1\right) \operatorname{atan}{\left(x + 1 \right)}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (x + 1)*atan(x + 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x+1)atan⁡(x+1)x)=−π2\lim_{x \to -\infty}\left(\frac{\left(x + 1\right) \operatorname{atan}{\left(x + 1 \right)}}{x}\right) = - \frac{\pi}{2}
Let's take the limit
so,
inclined asymptote equation on the left:
y=−πx2y = - \frac{\pi x}{2}
lim⁡x→∞((x+1)atan⁡(x+1)x)=π2\lim_{x \to \infty}\left(\frac{\left(x + 1\right) \operatorname{atan}{\left(x + 1 \right)}}{x}\right) = \frac{\pi}{2}
Let's take the limit
so,
inclined asymptote equation on the right:
y=πx2y = \frac{\pi x}{2}
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:
(x+1)atan⁡(x+1)=−(1−x)atan⁡(x−1)\left(x + 1\right) \operatorname{atan}{\left(x + 1 \right)} = - \left(1 - x\right) \operatorname{atan}{\left(x - 1 \right)}
- No
(x+1)atan⁡(x+1)=(1−x)atan⁡(x−1)\left(x + 1\right) \operatorname{atan}{\left(x + 1 \right)} = \left(1 - x\right) \operatorname{atan}{\left(x - 1 \right)}
- No
so, the function
not is
neither even, nor odd
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