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Plotting a function graph/ log(cot(x))^tan(x)

Graphing y = log(cot(x))^tan(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          tan(x)        
f(x) = log      (cot(x))
f(x)=log⁡(cot⁡(x))tan⁡(x)f{\left(x \right)} = \log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}} 
f = log(cot(x))^tan(x)
The graph of the function
02468-8-6-4-2-101002
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:
log⁡(cot⁡(x))tan⁡(x)=0\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π4x_{1} = \frac{\pi}{4}
Numerical solution
x1=−18.0641577581413x_{1} = -18.0641577581413
x2=−77.7544181763474x_{2} = -77.7544181763474
x3=−90.3207887907066x_{3} = -90.3207887907066
x4=69.9004365423729x_{4} = 69.9004365423729
x5=−99.7455667514759x_{5} = -99.7455667514759
x6=−33.7721210260903x_{6} = -33.7721210260903
x7=−55.7632696012188x_{7} = -55.7632696012188
x8=91.8915851175014x_{8} = 91.8915851175014
x9=−24.3473430653209x_{9} = -24.3473430653209
x10=10.2101761241668x_{10} = 10.2101761241668
x11=54.1924732744239x_{11} = 54.1924732744239
x12=−71.4712328691678x_{12} = -71.4712328691678
x13=60.4756585816035x_{13} = 60.4756585816035
x14=25.9181393921158x_{14} = 25.9181393921158
x15=−84.037603483527x_{15} = -84.037603483527
x16=76.1836218495525x_{16} = 76.1836218495525
x17=−40.0553063332699x_{17} = -40.0553063332699
x18=98.174770424681x_{18} = 98.174770424681
x19=3.92699081698724x_{19} = 3.92699081698724
x20=−62.0464549083984x_{20} = -62.0464549083984
x21=32.2013246992954x_{21} = 32.2013246992954
x22=38.484510006475x_{22} = 38.484510006475
x23=16.4933614313464x_{23} = 16.4933614313464
x24=−49.4800842940392x_{24} = -49.4800842940392
x25=−11.7809724509617x_{25} = -11.7809724509617
x26=−27.4889357189107x_{26} = -27.4889357189107
x27=82.4668071567321x_{27} = 82.4668071567321
x28=47.9092879672443x_{28} = 47.9092879672443
x29=−5.49778714378214x_{29} = -5.49778714378214
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(cot(x))^tan(x).
log⁡(cot⁡(0))tan⁡(0)\log{\left(\cot{\left(0 \right)} \right)}^{\tan{\left(0 \right)}}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞log⁡(cot⁡(x))tan⁡(x)y = \lim_{x \to -\infty} \log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞log⁡(cot⁡(x))tan⁡(x)y = \lim_{x \to \infty} \log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(cot(x))^tan(x), 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→−∞(log⁡(cot⁡(x))tan⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\log{\left(\cot{\left(x \right)} \right)}^{\tan{\left(x \right)}}}{x}\right)
True

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