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  • Graphing y =:
  • -3x+5
  • 3/2x^2-x^3
  • 3x^2-12x
  • |2x-4|+x
  • Integral of d{x}:
  • (x^2-1)e^x (x^2-1)e^x

  • Identical expressions

  • (x^ two - one)e^x
  • (x squared minus 1)e to the power of x
  • (x to the power of two minus one)e to the power of x
  • (x2-1)ex
  • x2-1ex
  • (x²-1)e^x
  • (x to the power of 2-1)e to the power of x
  • x^2-1e^x

  • Similar expressions

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

Graphing y = (x^2-1)e^x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       / 2    \  x
f(x) = \x  - 1/*E
f(x)=ex(x2−1)f{\left(x \right)} = e^{x} \left(x^{2} - 1\right) 
f = E^x*(x^2 - 1)
The graph of the function
02468-8-6-4-2-1010-25000002500000
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:
ex(x2−1)=0e^{x} \left(x^{2} - 1\right) = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−1x_{1} = -1
x2=1x_{2} = 1
Numerical solution
x1=−71.6025932625681x_{1} = -71.6025932625681
x2=−73.577910111863x_{2} = -73.577910111863
x3=−46.2180873563251x_{3} = -46.2180873563251
x4=−79.5129914542486x_{4} = -79.5129914542486
x5=−53.93969182993x_{5} = -53.93969182993
x6=−40.549966844823x_{6} = -40.549966844823
x7=−119.269715394308x_{7} = -119.269715394308
x8=−61.7576880804052x_{8} = -61.7576880804052
x9=−35.1157134585577x_{9} = -35.1157134585577
x10=−85.4590521969804x_{10} = -85.4590521969804
x11=−65.6883070820612x_{11} = -65.6883070820612
x12=−42.4219620932819x_{12} = -42.4219620932819
x13=−44.3126397955156x_{13} = -44.3126397955156
x14=−107.320766981846x_{14} = -107.320766981846
x15=−55.8874330100957x_{15} = -55.8874330100957
x16=−50.0625314136456x_{16} = -50.0625314136456
x17=−57.8401098081453x_{17} = -57.8401098081453
x18=1x_{18} = 1
x19=−111.302350381263x_{19} = -111.302350381263
x20=−81.4939382270426x_{20} = -81.4939382270426
x21=−105.330580740598x_{21} = -105.330580740598
x22=−117.277400270678x_{22} = -117.277400270678
x23=−67.6575690448132x_{23} = -67.6575690448132
x24=−103.34083441056x_{24} = -103.34083441056
x25=−95.3869000709892x_{25} = -95.3869000709892
x26=−77.5332552098772x_{26} = -77.5332552098772
x27=−89.4278853676496x_{27} = -89.4278853676496
x28=−99.3627857033581x_{28} = -99.3627857033581
x29=−109.311365356229x_{29} = -109.311365356229
x30=−69.6290775076557x_{30} = -69.6290775076557
x31=−113.293698637214x_{31} = -113.293698637214
x32=−83.4759897300653x_{32} = -83.4759897300653
x33=−38.7021606131911x_{33} = -38.7021606131911
x34=−1x_{34} = -1
x35=−101.351558330607x_{35} = -101.351558330607
x36=−48.1354367467417x_{36} = -48.1354367467417
x37=−115.285388562094x_{37} = -115.285388562094
x38=−87.4430422232197x_{38} = -87.4430422232197
x39=−36.8866033872636x_{39} = -36.8866033872636
x40=−121.262316938766x_{40} = -121.262316938766
x41=−97.3745529423273x_{41} = -97.3745529423273
x42=−91.4135149755732x_{42} = -91.4135149755732
x43=−93.3998711831007x_{43} = -93.3998711831007
x44=−75.5548493298981x_{44} = -75.5548493298981
x45=−51.9977149340079x_{45} = -51.9977149340079
x46=−59.7970469090044x_{46} = -59.7970469090044
x47=−63.7215707857911x_{47} = -63.7215707857911
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x^2 - 1)*E^x.
e0(−1+02)e^{0} \left(-1 + 0^{2}\right)
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
2xex+(x2−1)ex=02 x e^{x} + \left(x^{2} - 1\right) e^{x} = 0
Solve this equation
The roots of this equation
x1=−1+2x_{1} = -1 + \sqrt{2}
x2=−2−1x_{2} = - \sqrt{2} - 1
The values of the extrema at the points:
             /                 2\         ___ 
        ___  |     /       ___\ |  -1 + \/ 2  
(-1 + \/ 2, \-1 + \-1 + \/ 2 / /*e          )

             /                 2\         ___ 
        ___  |     /       ___\ |  -1 - \/ 2  
(-1 - \/ 2, \-1 + \-1 - \/ 2 / /*e          )


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=−1+2x_{1} = -1 + \sqrt{2}
Maxima of the function at points:
x1=−2−1x_{1} = - \sqrt{2} - 1
Decreasing at intervals
(−∞,−2−1]∪[−1+2,∞)\left(-\infty, - \sqrt{2} - 1\right] \cup \left[-1 + \sqrt{2}, \infty\right)
Increasing at intervals
[−2−1,−1+2]\left[- \sqrt{2} - 1, -1 + \sqrt{2}\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
(x2+4x+1)ex=0\left(x^{2} + 4 x + 1\right) e^{x} = 0
Solve this equation
The roots of this equation
x1=−2−3x_{1} = -2 - \sqrt{3}
x2=−2+3x_{2} = -2 + \sqrt{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
(−∞,−2−3]∪[−2+3,∞)\left(-\infty, -2 - \sqrt{3}\right] \cup \left[-2 + \sqrt{3}, \infty\right)
Convex at the intervals
[−2−3,−2+3]\left[-2 - \sqrt{3}, -2 + \sqrt{3}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(ex(x2−1))=0\lim_{x \to -\infty}\left(e^{x} \left(x^{2} - 1\right)\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(ex(x2−1))=∞\lim_{x \to \infty}\left(e^{x} \left(x^{2} - 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^2 - 1)*E^x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x2−1)exx)=0\lim_{x \to -\infty}\left(\frac{\left(x^{2} - 1\right) e^{x}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞((x2−1)exx)=∞\lim_{x \to \infty}\left(\frac{\left(x^{2} - 1\right) e^{x}}{x}\right) = \infty
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
ex(x2−1)=(x2−1)e−xe^{x} \left(x^{2} - 1\right) = \left(x^{2} - 1\right) e^{- x}
- No
ex(x2−1)=−(x2−1)e−xe^{x} \left(x^{2} - 1\right) = - \left(x^{2} - 1\right) e^{- x}
- No
so, the function
not is
neither even, nor odd
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