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  • How to use it?

  • Graphing y =:
  • -x^2-2x+15
  • ((x-1)/(x+1))^3
  • (x-1)^2*(x+2)
  • 2x^3-3x^2-16

  • Identical expressions

  • x*exp(one / two -x^ two)
  • x multiply by exponent of (1 divide by 2 minus x squared )
  • x multiply by exponent of (one divide by two minus x to the power of two)
  • x*exp(1/2-x2)
  • x*exp1/2-x2
  • x*exp(1/2-x²)
  • x*exp(1/2-x to the power of 2)
  • xexp(1/2-x^2)
  • xexp(1/2-x2)
  • xexp1/2-x2
  • xexp1/2-x^2
  • x*exp(1 divide by 2-x^2)

  • Similar expressions

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

Graphing y = x*exp(1/2-x^2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          1    2
          - - x 
          2     
f(x) = x*e
f(x)=xe12−x2f{\left(x \right)} = x e^{\frac{1}{2} - x^{2}} 
f = x*exp(1/2 - x^2)
The graph of the function
02468-8-6-4-2-10102-2
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:
xe12−x2=0x e^{\frac{1}{2} - x^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=72.3905669642191x_{1} = 72.3905669642191
x2=−96.1022227575016x_{2} = -96.1022227575016
x3=−64.1532478546075x_{3} = -64.1532478546075
x4=−98.0051017674524x_{4} = -98.0051017674524
x5=−14.6873692419514x_{5} = -14.6873692419514
x6=−48.2041698444887x_{6} = -48.2041698444887
x7=38.5149532926537x_{7} = 38.5149532926537
x8=−94.1043957123178x_{8} = -94.1043957123178
x9=28.6078690007847x_{9} = 28.6078690007847
x10=−32.3055705254499x_{10} = -32.3055705254499
x11=−16.6040954166054x_{11} = -16.6040954166054
x12=−18.5386580960287x_{12} = -18.5386580960287
x13=32.5638726903793x_{13} = 32.5638726903793
x14=40.5018595731864x_{14} = 40.5018595731864
x15=−60.1634415856681x_{15} = -60.1634415856681
x16=7.73387960889725x_{16} = 7.73387960889725
x17=−38.2576235616253x_{17} = -38.2576235616253
x18=−7.49944089361232x_{18} = -7.49944089361232
x19=−56.1750861644869x_{19} = -56.1750861644869
x20=14.9488147933141x_{20} = 14.9488147933141
x21=−44.2226547762937x_{21} = -44.2226547762937
x22=16.8656081194959x_{22} = 16.8656081194959
x23=24.6661073409247x_{23} = 24.6661073409247
x24=90.3625649128685x_{24} = 90.3625649128685
x25=−86.1140964733667x_{25} = -86.1140964733667
x26=52.4442259550818x_{26} = 52.4442259550818
x27=48.4602702583442x_{27} = 48.4602702583442
x28=−42.2332088562321x_{28} = -42.2332088562321
x29=11.204731998154x_{29} = 11.204731998154
x30=94.3577924641268x_{30} = 94.3577924641268
x31=100.254985258461x_{31} = 100.254985258461
x32=−28.3487957849917x_{32} = -28.3487957849917
x33=−84.1168098950623x_{33} = -84.1168098950623
x34=−90.1090308486596x_{34} = -90.1090308486596
x35=26.6348104846935x_{35} = 26.6348104846935
x36=−10.9467267431996x_{36} = -10.9467267431996
x37=46.4693271688763x_{37} = 46.4693271688763
x38=−12.7968091698462x_{38} = -12.7968091698462
x39=42.4899960942614x_{39} = 42.4899960942614
x40=−62.1581808370582x_{40} = -62.1581808370582
x41=−50.1960306485386x_{41} = -50.1960306485386
x42=96.3555548022329x_{42} = 96.3555548022329
x43=78.3798045164051x_{43} = 78.3798045164051
x44=60.4185026243398x_{44} = 60.4185026243398
x45=6.25999679126562x_{45} = 6.25999679126562
x46=88.3651131646469x_{46} = 88.3651131646469
x47=0x_{47} = 0
x48=−82.1196554412931x_{48} = -82.1196554412931
x49=−52.1885145543938x_{49} = -52.1885145543938
x50=20.7467570641594x_{50} = 20.7467570641594
x51=−24.4061660554474x_{51} = -24.4061660554474
x52=56.4304534331944x_{52} = 56.4304534331944
x53=84.3705719720026x_{53} = 84.3705719720026
x54=−72.1362492686553x_{54} = -72.1362492686553
x55=−100.005001053706x_{55} = -100.005001053706
x56=80.3765739119403x_{56} = 80.3765739119403
x57=−68.1442499509512x_{57} = -68.1442499509512
x58=58.4242736475066x_{58} = 58.4242736475066
x59=36.5294784207713x_{59} = 36.5294784207713
x60=62.4131011494136x_{60} = 62.4131011494136
x61=−76.129088844583x_{61} = -76.129088844583
x62=30.5844367675871x_{62} = 30.5844367675871
x63=68.3987904608748x_{63} = 68.3987904608748
x64=66.4032735361933x_{64} = 66.4032735361933
x65=−74.132572514098x_{65} = -74.132572514098
x66=−54.1815527450525x_{66} = -54.1815527450525
x67=92.3601269622633x_{67} = 92.3601269622633
x68=54.4370869389189x_{68} = 54.4370869389189
x69=74.3867866673358x_{69} = 74.3867866673358
x70=−6.0660448195699x_{70} = -6.0660448195699
x71=−9.16348321175021x_{71} = -9.16348321175021
x72=18.7999073667175x_{72} = 18.7999073667175
x73=−78.125783449603x_{73} = -78.125783449603
x74=−88.1115061822663x_{74} = -88.1115061822663
x75=76.3832042269081x_{75} = 76.3832042269081
x76=−40.244810563854x_{76} = -40.244810563854
x77=−22.4425007185933x_{77} = -22.4425007185933
x78=82.3735001099732x_{78} = 82.3735001099732
x79=−26.3753139220651x_{79} = -26.3753139220651
x80=44.4791976169932x_{80} = 44.4791976169932
x81=64.408034863895x_{81} = 64.408034863895
x82=−20.4859087535338x_{82} = -20.4859087535338
x83=50.4519304520445x_{83} = 50.4519304520445
x84=−34.2877276681532x_{84} = -34.2877276681532
x85=−46.2130128196551x_{85} = -46.2130128196551
x86=9.41469724573827x_{86} = 9.41469724573827
x87=34.5456823438052x_{87} = 34.5456823438052
x88=−36.2718473606153x_{88} = -36.2718473606153
x89=−30.3257619544685x_{89} = -30.3257619544685
x90=86.3677793889685x_{90} = 86.3677793889685
x91=−58.1690638483732x_{91} = -58.1690638483732
x92=13.0574223701178x_{92} = 13.0574223701178
x93=70.3945619595659x_{93} = 70.3945619595659
x94=−80.122642994584x_{94} = -80.122642994584
x95=22.7028991452193x_{95} = 22.7028991452193
x96=−66.1486129602884x_{96} = -66.1486129602884
x97=−70.1401356046363x_{97} = -70.1401356046363
x98=98.2550892569467x_{98} = 98.2550892569467
x99=−92.1066629479684x_{99} = -92.1066629479684
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*exp(1/2 - x^2).
0e12−020 e^{\frac{1}{2} - 0^{2}}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
−2x2e12−x2+e12−x2=0- 2 x^{2} e^{\frac{1}{2} - x^{2}} + e^{\frac{1}{2} - x^{2}} = 0
Solve this equation
The roots of this equation
x1=−22x_{1} = - \frac{\sqrt{2}}{2}
x2=22x_{2} = \frac{\sqrt{2}}{2}
The values of the extrema at the points:
    ___      ___  
 -\/ 2    -\/ 2   
(-------, -------)
    2        2    

   ___    ___ 
 \/ 2   \/ 2  
(-----, -----)
   2      2


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=−22x_{1} = - \frac{\sqrt{2}}{2}
Maxima of the function at points:
x1=22x_{1} = \frac{\sqrt{2}}{2}
Decreasing at intervals
[−22,22]\left[- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right]
Increasing at intervals
(−∞,−22]∪[22,∞)\left(-\infty, - \frac{\sqrt{2}}{2}\right] \cup \left[\frac{\sqrt{2}}{2}, \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
2x(2x2−3)e12−x2=02 x \left(2 x^{2} - 3\right) e^{\frac{1}{2} - x^{2}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−62x_{2} = - \frac{\sqrt{6}}{2}
x3=62x_{3} = \frac{\sqrt{6}}{2}

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
[−62,0]∪[62,∞)\left[- \frac{\sqrt{6}}{2}, 0\right] \cup \left[\frac{\sqrt{6}}{2}, \infty\right)
Convex at the intervals
(−∞,−62]∪[0,62]\left(-\infty, - \frac{\sqrt{6}}{2}\right] \cup \left[0, \frac{\sqrt{6}}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xe12−x2)=0\lim_{x \to -\infty}\left(x e^{\frac{1}{2} - x^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(xe12−x2)=0\lim_{x \to \infty}\left(x e^{\frac{1}{2} - 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 x*exp(1/2 - x^2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞e12−x2=0\lim_{x \to -\infty} e^{\frac{1}{2} - x^{2}} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞e12−x2=0\lim_{x \to \infty} e^{\frac{1}{2} - x^{2}} = 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:
xe12−x2=−xe12−x2x e^{\frac{1}{2} - x^{2}} = - x e^{\frac{1}{2} - x^{2}}
- No
xe12−x2=xe12−x2x e^{\frac{1}{2} - x^{2}} = x e^{\frac{1}{2} - x^{2}}
- Yes
so, the function
is
odd
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