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  • How to use it?

  • Graphing y =:
  • -x^2-2x+1
  • (x+1)(x-2)^2
  • 6x-2x^2
  • 9^(1/(x-3))

  • Identical expressions

  • (x+ one)/(x*x+ one)^ two
  • (x plus 1) divide by (x multiply by x plus 1) squared
  • (x plus one) divide by (x multiply by x plus one) to the power of two
  • (x+1)/(x*x+1)2
  • x+1/x*x+12
  • (x+1)/(x*x+1)²
  • (x+1)/(x*x+1) to the power of 2
  • (x+1)/(xx+1)^2
  • (x+1)/(xx+1)2
  • x+1/xx+12
  • x+1/xx+1^2
  • (x+1) divide by (x*x+1)^2

  • Similar expressions

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

Graphing y = (x+1)/(x*x+1)^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
         x + 1   
f(x) = ----------
                2
       (x*x + 1)
f(x)=x+1(xx+1)2f{\left(x \right)} = \frac{x + 1}{\left(x x + 1\right)^{2}} 
f = (x + 1)/(x*x + 1)^2
The graph of the function
02468-8-6-4-2-10102-1
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:
x+1(xx+1)2=0\frac{x + 1}{\left(x x + 1\right)^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−1x_{1} = -1
Numerical solution
x1=33834.2285454912x_{1} = 33834.2285454912
x2=−26358.1255649569x_{2} = -26358.1255649569
x3=−42459.8980123181x_{3} = -42459.8980123181
x4=30444.4033541377x_{4} = 30444.4033541377
x5=40614.1727065763x_{5} = 40614.1727065763
x6=−28900.3073791784x_{6} = -28900.3073791784
x7=31291.8477670117x_{7} = 31291.8477670117
x8=−15344.4829215851x_{8} = -15344.4829215851
x9=−39069.8638512438x_{9} = -39069.8638512438
x10=−41612.3835710987x_{10} = -41612.3835710987
x11=42309.2029299978x_{11} = 42309.2029299978
x12=−11111.3201545584x_{12} = -11111.3201545584
x13=32986.7609668844x_{13} = 32986.7609668844
x14=22817.9297180831x_{14} = 22817.9297180831
x15=10113.508304835x_{15} = 10113.508304835
x16=27902.1276363793x_{16} = 27902.1276363793
x17=−17885.5101822959x_{17} = -17885.5101822959
x18=−9419.33329706892x_{18} = -9419.33329706892
x19=−10265.1883488199x_{19} = -10265.1883488199
x20=36376.6691166606x_{20} = 36376.6691166606
x21=−20426.9904729273x_{21} = -20426.9904729273
x22=38071.6568546614x_{22} = 38071.6568546614
x23=−25510.7599982124x_{23} = -25510.7599982124
x24=−34832.4270840261x_{24} = -34832.4270840261
x25=−22121.4838361446x_{25} = -22121.4838361446
x26=23665.2540647425x_{26} = 23665.2540647425
x27=−33984.9574174783x_{27} = -33984.9574174783
x28=20276.0946617741x_{28} = 20276.0946617741
x29=16887.4298531687x_{29} = 16887.4298531687
x30=−32290.0395295242x_{30} = -32290.0395295242
x31=37224.160496723x_{31} = 37224.160496723
x32=35529.1830671647x_{32} = 35529.1830671647
x33=16040.3768972116x_{33} = 16040.3768972116
x34=29596.9679855437x_{34} = 29596.9679855437
x35=−37374.873078274x_{35} = -37374.873078274
x36=−35679.9032124213x_{36} = -35679.9032124213
x37=21123.3472938935x_{37} = 21123.3472938935
x38=−40764.8728987272x_{38} = -40764.8728987272
x39=14346.4617481587x_{39} = 14346.4617481587
x40=15193.3840611629x_{40} = 15193.3840611629
x41=−33137.4947143549x_{41} = -33137.4947143549
x42=34681.7027350514x_{42} = 34681.7027350514
x43=−39917.3662375731x_{43} = -39917.3662375731
x44=38919.1578682278x_{44} = 38919.1578682278
x45=25359.9581060261x_{45} = 25359.9581060261
x46=21970.6266224823x_{46} = 21970.6266224823
x47=9267.45404568973x_{47} = 9267.45404568973
x48=−36527.3853481304x_{48} = -36527.3853481304
x49=39766.6632424023x_{49} = 39766.6632424023
x50=−21274.2226481452x_{50} = -21274.2226481452
x51=28749.5424513273x_{51} = 28749.5424513273
x52=18581.6838276728x_{52} = 18581.6838276728
x53=11806.2655234967x_{53} = 11806.2655234967
x54=−29747.7255943491x_{54} = -29747.7255943491
x55=−18732.6292450164x_{55} = -18732.6292450164
x56=13499.622908825x_{56} = 13499.622908825
x57=19428.8721576817x_{57} = 19428.8721576817
x58=−31442.592478417x_{58} = -31442.592478417
x59=26207.3342685621x_{59} = 26207.3342685621
x60=−12804.1871325825x_{60} = -12804.1871325825
x61=−27205.5062138373x_{61} = -27205.5062138373
x62=−17038.4405090108x_{62} = -17038.4405090108
x63=−28052.9005589566x_{63} = -28052.9005589566
x64=24512.5974895349x_{64} = 24512.5974895349
x65=27054.7245350401x_{65} = 27054.7245350401
x66=−11957.667829554x_{66} = -11957.667829554
x67=−22968.7707704751x_{67} = -22968.7707704751
x68=41461.6860120789x_{68} = 41461.6860120789
x69=−30595.1542455215x_{69} = -30595.1542455215
x70=−38222.3660269625x_{70} = -38222.3660269625
x71=−14497.6168493628x_{71} = -14497.6168493628
x72=12652.8838720334x_{72} = 12652.8838720334
x73=10959.7950148984x_{73} = 10959.7950148984
x74=−16191.4281723745x_{74} = -16191.4281723745
x75=−19579.7911539546x_{75} = -19579.7911539546
x76=−13650.8451413168x_{76} = -13650.8451413168
x77=32139.3005164984x_{77} = 32139.3005164984
x78=17734.5344763108x_{78} = 17734.5344763108
x79=−23816.0806573264x_{79} = -23816.0806573264
x80=−24663.4110930219x_{80} = -24663.4110930219
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x + 1)/(x*x + 1)^2.
1(0⋅0+1)2\frac{1}{\left(0 \cdot 0 + 1\right)^{2}}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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
−4x(x+1)(xx+1)3+1(xx+1)2=0- \frac{4 x \left(x + 1\right)}{\left(x x + 1\right)^{3}} + \frac{1}{\left(x x + 1\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−23+73x_{1} = - \frac{2}{3} + \frac{\sqrt{7}}{3}
x2=−73−23x_{2} = - \frac{\sqrt{7}}{3} - \frac{2}{3}
The values of the extrema at the points:
                          ___       
                    1   \/ 7        
         ___        - + -----       
   2   \/ 7         3     3         
(- - + -----, ---------------------)
   3     3                        2 
              /                 2\  
              |    /        ___\ |  
              |    |  2   \/ 7 | |  
              |1 + |- - + -----| |  
              \    \  3     3  / /  

                          ___       
                    1   \/ 7        
         ___        - - -----       
   2   \/ 7         3     3         
(- - - -----, ---------------------)
   3     3                        2 
              /                 2\  
              |    /        ___\ |  
              |    |  2   \/ 7 | |  
              |1 + |- - - -----| |  
              \    \  3     3  / /


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=−73−23x_{1} = - \frac{\sqrt{7}}{3} - \frac{2}{3}
Maxima of the function at points:
x1=−23+73x_{1} = - \frac{2}{3} + \frac{\sqrt{7}}{3}
Decreasing at intervals
[−73−23,−23+73]\left[- \frac{\sqrt{7}}{3} - \frac{2}{3}, - \frac{2}{3} + \frac{\sqrt{7}}{3}\right]
Increasing at intervals
(−∞,−73−23]∪[−23+73,∞)\left(-\infty, - \frac{\sqrt{7}}{3} - \frac{2}{3}\right] \cup \left[- \frac{2}{3} + \frac{\sqrt{7}}{3}, \infty\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
4(−2x+(x+1)(6x2x2+1−1))(x2+1)3=0\frac{4 \left(- 2 x + \left(x + 1\right) \left(\frac{6 x^{2}}{x^{2} + 1} - 1\right)\right)}{\left(x^{2} + 1\right)^{3}} = 0
Solve this equation
The roots of this equation
x1=−59−20627+2303i333−522720627+2303i33x_{1} = - \frac{5}{9} - \frac{\sqrt[3]{\frac{206}{27} + \frac{2 \sqrt{303} i}{3}}}{3} - \frac{52}{27 \sqrt[3]{\frac{206}{27} + \frac{2 \sqrt{303} i}{3}}}

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
[−413cos⁡(atan⁡(9303103)3)9−59,∞)\left[- \frac{4 \sqrt{13} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{303}}{103} \right)}}{3} \right)}}{9} - \frac{5}{9}, \infty\right)
Convex at the intervals
(−∞,−413cos⁡(atan⁡(9303103)3)9−59]\left(-\infty, - \frac{4 \sqrt{13} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{303}}{103} \right)}}{3} \right)}}{9} - \frac{5}{9}\right]
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(xx+1)2)=0\lim_{x \to -\infty}\left(\frac{x + 1}{\left(x x + 1\right)^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(x+1(xx+1)2)=0\lim_{x \to \infty}\left(\frac{x + 1}{\left(x x + 1\right)^{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 (x + 1)/(x*x + 1)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x+1x(xx+1)2)=0\lim_{x \to -\infty}\left(\frac{x + 1}{x \left(x x + 1\right)^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(x+1x(xx+1)2)=0\lim_{x \to \infty}\left(\frac{x + 1}{x \left(x x + 1\right)^{2}}\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:
x+1(xx+1)2=1−x(x2+1)2\frac{x + 1}{\left(x x + 1\right)^{2}} = \frac{1 - x}{\left(x^{2} + 1\right)^{2}}
- No
x+1(xx+1)2=−1−x(x2+1)2\frac{x + 1}{\left(x x + 1\right)^{2}} = - \frac{1 - x}{\left(x^{2} + 1\right)^{2}}
- No
so, the function
not is
neither even, nor odd
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