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Plotting a function graph/ ctg((x+1)/2)

Graphing y = ctg((x+1)/2)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          /x + 1\
f(x) = cot|-----|
          \  2  /
f(x)=cot⁡(x+12)f{\left(x \right)} = \cot{\left(\frac{x + 1}{2} \right)} 
f = cot((x + 1)/2)
The graph of the function
02468-8-6-4-2-1010-1000500
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:
cot⁡(x+12)=0\cot{\left(\frac{x + 1}{2} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−1+πx_{1} = -1 + \pi
Numerical solution
x1=77.5398163397448x_{1} = 77.5398163397448
x2=96.3893722612836x_{2} = 96.3893722612836
x3=−54.4070751110265x_{3} = -54.4070751110265
x4=71.2566310325652x_{4} = 71.2566310325652
x5=−60.6902604182061x_{5} = -60.6902604182061
x6=39.8407044966673x_{6} = 39.8407044966673
x7=−85.8230016469244x_{7} = -85.8230016469244
x8=−79.5398163397448x_{8} = -79.5398163397448
x9=102.672557568463x_{9} = 102.672557568463
x10=−10.4247779607694x_{10} = -10.4247779607694
x11=−35.5575191894877x_{11} = -35.5575191894877
x12=90.106186954104x_{12} = 90.106186954104
x13=−48.1238898038469x_{13} = -48.1238898038469
x14=64.9734457253857x_{14} = 64.9734457253857
x15=−73.2566310325652x_{15} = -73.2566310325652
x16=−29.2743338823081x_{16} = -29.2743338823081
x17=58.6902604182061x_{17} = 58.6902604182061
x18=8.42477796076938x_{18} = 8.42477796076938
x19=−98.3893722612836x_{19} = -98.3893722612836
x20=52.4070751110265x_{20} = 52.4070751110265
x21=−22.9911485751286x_{21} = -22.9911485751286
x22=46.1238898038469x_{22} = 46.1238898038469
x23=2.14159265358979x_{23} = 2.14159265358979
x24=−66.9734457253857x_{24} = -66.9734457253857
x25=−16.707963267949x_{25} = -16.707963267949
x26=−92.106186954104x_{26} = -92.106186954104
x27=−41.8407044966673x_{27} = -41.8407044966673
x28=83.8230016469244x_{28} = 83.8230016469244
x29=27.2743338823081x_{29} = 27.2743338823081
x30=14.707963267949x_{30} = 14.707963267949
x31=20.9911485751286x_{31} = 20.9911485751286
x32=−4.14159265358979x_{32} = -4.14159265358979
x33=33.5575191894877x_{33} = 33.5575191894877
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot((x + 1)/2).
cot⁡(12)\cot{\left(\frac{1}{2} \right)}
The result:
f(0)=cot⁡(12)f{\left(0 \right)} = \cot{\left(\frac{1}{2} \right)}
The point:
(0, cot(1/2))
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
−cot⁡2(x+12)2−12=0- \frac{\cot^{2}{\left(\frac{x + 1}{2} \right)}}{2} - \frac{1}{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
(cot⁡2(x+12)+1)cot⁡(x+12)2=0\frac{\left(\cot^{2}{\left(\frac{x + 1}{2} \right)} + 1\right) \cot{\left(\frac{x + 1}{2} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=−1+πx_{1} = -1 + \pi

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+π]\left(-\infty, -1 + \pi\right]
Convex at the intervals
[−1+π,∞)\left[-1 + \pi, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞cot⁡(x+12)=−cot⁡(∞)\lim_{x \to -\infty} \cot{\left(\frac{x + 1}{2} \right)} = - \cot{\left(\infty \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−cot⁡(∞)y = - \cot{\left(\infty \right)}
lim⁡x→∞cot⁡(x+12)=cot⁡(∞)\lim_{x \to \infty} \cot{\left(\frac{x + 1}{2} \right)} = \cot{\left(\infty \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=cot⁡(∞)y = \cot{\left(\infty \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot((x + 1)/2), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(cot⁡(x+12)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(\frac{x + 1}{2} \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(x+12)x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(\frac{x + 1}{2} \right)}}{x}\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:
cot⁡(x+12)=−cot⁡(x2−12)\cot{\left(\frac{x + 1}{2} \right)} = - \cot{\left(\frac{x}{2} - \frac{1}{2} \right)}
- No
cot⁡(x+12)=cot⁡(x2−12)\cot{\left(\frac{x + 1}{2} \right)} = \cot{\left(\frac{x}{2} - \frac{1}{2} \right)}
- No
so, the function
not is
neither even, nor odd
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