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

Graphing y = (x-1)e^x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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                x
f(x) = (x - 1)*e
f(x)=(x−1)exf{\left(x \right)} = \left(x - 1\right) e^{x} 
f = (x - 1*1)*E^x
The graph of the function
0-100-90-80-70-60-50-40-30-20-1010400000-200000
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)ex=0\left(x - 1\right) e^{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
Numerical solution
x1=−109.087371742331x_{1} = -109.087371742331
x2=−99.1120495157731x_{2} = -99.1120495157731
x3=−77.1931311289629x_{3} = -77.1931311289629
x4=−53.3821676071309x_{4} = -53.3821676071309
x5=−95.1235868161767x_{5} = -95.1235868161767
x6=−45.5083552648416x_{6} = -45.5083552648416
x7=−111.08303446753x_{7} = -111.08303446753
x8=−67.2515753571383x_{8} = -67.2515753571383
x9=−51.4086841814429x_{9} = -51.4086841814429
x10=−115.074865014488x_{10} = -115.074865014488
x11=−91.1362942896831x_{11} = -91.1362942896831
x12=−59.316486753355x_{12} = -59.316486753355
x13=−47.4711655449634x_{13} = -47.4711655449634
x14=−113.078868899778x_{14} = -113.078868899778
x15=−101.106670133692x_{15} = -101.106670133692
x16=−69.2382302560517x_{16} = -69.2382302560517
x17=−49.4381699084522x_{17} = -49.4381699084522
x18=−97.1176822742156x_{18} = -97.1176822742156
x19=−105.096605847552x_{19} = -105.096605847552
x20=−61.2982393476586x_{20} = -61.2982393476586
x21=−121.063734292694x_{21} = -121.063734292694
x22=1x_{22} = 1
x23=−71.2257989645248x_{23} = -71.2257989645248
x24=−83.1660166222937x_{24} = -83.1660166222937
x25=−93.1297833837852x_{25} = -93.1297833837852
x26=−63.2814467335924x_{26} = -63.2814467335924
x27=−73.2141900449367x_{27} = -73.2141900449367
x28=−33.8971886855811x_{28} = -33.8971886855811
x29=−41.5991101904548x_{29} = -41.5991101904548
x30=−65.2659399232894x_{30} = -65.2659399232894
x31=−87.1503604017549x_{31} = -87.1503604017549
x32=−81.1745282419576x_{32} = -81.1745282419576
x33=−75.2033239479075x_{33} = -75.2033239479075
x34=−35.8006485741225x_{34} = -35.8006485741225
x35=−89.1431441899768x_{35} = -89.1431441899768
x36=−117.071013554438x_{36} = -117.071013554438
x37=−79.1835505142898x_{37} = -79.1835505142898
x38=−32.0182140925185x_{38} = -32.0182140925185
x39=−37.7215440170094x_{39} = -37.7215440170094
x40=−119.06730595755x_{40} = -119.06730595755
x41=−85.157973273941x_{41} = -85.157973273941
x42=−55.3581866464466x_{42} = -55.3581866464466
x43=−107.091891597578x_{43} = -107.091891597578
x44=−39.6553752443623x_{44} = -39.6553752443623
x45=−43.550618994199x_{45} = -43.550618994199
x46=−103.101527351786x_{46} = -103.101527351786
x47=−57.336389337426x_{47} = -57.336389337426
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (x - 1*1)*E^x.
((−1)1+0)e0\left(\left(-1\right) 1 + 0\right) e^{0}
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
(x−1)ex+ex=0\left(x - 1\right) e^{x} + e^{x} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
The values of the extrema at the points:
(0, -1*1)


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=0x_{1} = 0
The function has no maxima
Decreasing at intervals
[0,∞)\left[0, \infty\right)
Increasing at intervals
(−∞,0]\left(-\infty, 0\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
(x+1)ex=0\left(x + 1\right) e^{x} = 0
Solve this equation
The roots of this equation
x1=−1x_{1} = -1

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
[−1,∞)\left[-1, \infty\right)
Convex at the intervals
(−∞,−1]\left(-\infty, -1\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)ex)=0\lim_{x \to -\infty}\left(\left(x - 1\right) e^{x}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞((x−1)ex)=∞\lim_{x \to \infty}\left(\left(x - 1\right) e^{x}\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*1)*E^x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞((x−1)exx)=0\lim_{x \to -\infty}\left(\frac{\left(x - 1\right) e^{x}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞((x−1)exx)=∞\lim_{x \to \infty}\left(\frac{\left(x - 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:
(x−1)ex=(−x−1)e−x\left(x - 1\right) e^{x} = \left(- x - 1\right) e^{- x}
- No
(x−1)ex=−(−x−1)e−x\left(x - 1\right) e^{x} = - \left(- x - 1\right) e^{- x}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = (x-1)e^x
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