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

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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               x
       1 - (-1) 
f(x) = ---------
        2*x + 1
f(x)=1−(−1)x2x+1f{\left(x \right)} = \frac{1 - \left(-1\right)^{x}}{2 x + 1} 
f = (1 - (-1)^x)/(2*x + 1)
The graph of the function
-0.010-0.008-0.006-0.004-0.0020.0100.0000.0020.0040.0060.0080.00
The domain of the function
The points at which the function is not precisely defined:
x1=−0.5x_{1} = -0.5
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:
1−(−1)x2x+1=0\frac{1 - \left(-1\right)^{x}}{2 x + 1} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=−42x_{1} = -42
x2=82x_{2} = 82
x3=−66x_{3} = -66
x4=76x_{4} = 76
x5=4x_{5} = 4
x6=−30x_{6} = -30
x7=16x_{7} = 16
x8=−22x_{8} = -22
x9=−76x_{9} = -76
x10=88x_{10} = 88
x11=62x_{11} = 62
x12=8x_{12} = 8
x13=66x_{13} = 66
x14=92x_{14} = 92
x15=−62x_{15} = -62
x16=94x_{16} = 94
x17=26x_{17} = 26
x18=46x_{18} = 46
x19=22x_{19} = 22
x20=−78x_{20} = -78
x21=−48x_{21} = -48
x22=100x_{22} = 100
x23=−60x_{23} = -60
x24=−90x_{24} = -90
x25=−6x_{25} = -6
x26=12x_{26} = 12
x27=32x_{27} = 32
x28=30x_{28} = 30
x29=−34x_{29} = -34
x30=−20x_{30} = -20
x31=−52x_{31} = -52
x32=−96x_{32} = -96
x33=−64x_{33} = -64
x34=68x_{34} = 68
x35=72x_{35} = 72
x36=50x_{36} = 50
x37=−82x_{37} = -82
x38=−72x_{38} = -72
x39=−24x_{39} = -24
x40=52x_{40} = 52
x41=−10x_{41} = -10
x42=−8x_{42} = -8
x43=10x_{43} = 10
x44=84x_{44} = 84
x45=−28x_{45} = -28
x46=−84x_{46} = -84
x47=24x_{47} = 24
x48=64x_{48} = 64
x49=36x_{49} = 36
x50=−88x_{50} = -88
x51=−2x_{51} = -2
x52=14x_{52} = 14
x53=−38x_{53} = -38
x54=60x_{54} = 60
x55=56x_{55} = 56
x56=−86x_{56} = -86
x57=40x_{57} = 40
x58=−46x_{58} = -46
x59=−74x_{59} = -74
x60=−26x_{60} = -26
x61=−32x_{61} = -32
x62=58x_{62} = 58
x63=70x_{63} = 70
x64=−70x_{64} = -70
x65=18x_{65} = 18
x66=−68x_{66} = -68
x67=−14x_{67} = -14
x68=44x_{68} = 44
x69=6x_{69} = 6
x70=−54x_{70} = -54
x71=−58x_{71} = -58
x72=90x_{72} = 90
x73=98x_{73} = 98
x74=−92x_{74} = -92
x75=−94x_{75} = -94
x76=80x_{76} = 80
x77=86x_{77} = 86
x78=−44x_{78} = -44
x79=−18x_{79} = -18
x80=0x_{80} = 0
x81=48x_{81} = 48
x82=96x_{82} = 96
x83=−80x_{83} = -80
x84=−16x_{84} = -16
x85=20x_{85} = 20
x86=−12x_{86} = -12
x87=−56x_{87} = -56
x88=34x_{88} = 34
x89=−40x_{89} = -40
x90=−50x_{90} = -50
x91=28x_{91} = 28
x92=2x_{92} = 2
x93=−36x_{93} = -36
x94=74x_{94} = 74
x95=42x_{95} = 42
x96=54x_{96} = 54
x97=−98x_{97} = -98
x98=−4x_{98} = -4
x99=−100x_{99} = -100
x100=78x_{100} = 78
x101=38x_{101} = 38
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (1 - (-1)^x)/(2*x + 1).
1−(−1)00⋅2+1\frac{1 - \left(-1\right)^{0}}{0 \cdot 2 + 1}
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
−(−1)xiπ2x+1−2(1−(−1)x)(2x+1)2=0- \frac{\left(-1\right)^{x} i \pi}{2 x + 1} - \frac{2 \left(1 - \left(-1\right)^{x}\right)}{\left(2 x + 1\right)^{2}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
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
(−1)xπ2+4(−1)xiπ2x+1−8((−1)x−1)(2x+1)22x+1=0\frac{\left(-1\right)^{x} \pi^{2} + \frac{4 \left(-1\right)^{x} i \pi}{2 x + 1} - \frac{8 \left(\left(-1\right)^{x} - 1\right)}{\left(2 x + 1\right)^{2}}}{2 x + 1} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=−0.5x_{1} = -0.5
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Limit on the left could not be calculated
lim⁡x→−∞(1−(−1)x2x+1)\lim_{x \to -\infty}\left(\frac{1 - \left(-1\right)^{x}}{2 x + 1}\right)
Limit on the right could not be calculated
lim⁡x→∞(1−(−1)x2x+1)\lim_{x \to \infty}\left(\frac{1 - \left(-1\right)^{x}}{2 x + 1}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1 - (-1)^x)/(2*x + 1), divided by x at x->+oo and x ->-oo
Limit on the left could not be calculated
lim⁡x→−∞(1−(−1)xx(2x+1))\lim_{x \to -\infty}\left(\frac{1 - \left(-1\right)^{x}}{x \left(2 x + 1\right)}\right)
Limit on the right could not be calculated
lim⁡x→∞(1−(−1)xx(2x+1))\lim_{x \to \infty}\left(\frac{1 - \left(-1\right)^{x}}{x \left(2 x + 1\right)}\right)
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:
1−(−1)x2x+1=1−(−1)−x1−2x\frac{1 - \left(-1\right)^{x}}{2 x + 1} = \frac{1 - \left(-1\right)^{- x}}{1 - 2 x}
- No
1−(−1)x2x+1=−1−(−1)−x1−2x\frac{1 - \left(-1\right)^{x}}{2 x + 1} = - \frac{1 - \left(-1\right)^{- x}}{1 - 2 x}
- No
so, the function
not is
neither even, nor odd
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