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  • Graphing y =:
  • x^3+3x^2-9x+15
  • -x^2+6x-8
  • x^2-6x+10
  • (x^2+5)/(x-2)

  • Identical expressions

  • (tgx*arctg(one /x- two))/x
  • (tgx multiply by arctg(1 divide by x minus 2)) divide by x
  • (tgx multiply by arctg(one divide by x minus two)) divide by x
  • (tgxarctg(1/x-2))/x
  • tgxarctg1/x-2/x
  • (tgx*arctg(1 divide by x-2)) divide by x

  • Similar expressions

  • (tgx*arctg(1/x+2))/x
Plotting a function graph/ (tgx*arctg(1/x-2))/x

Graphing y = (tgx*arctg(1/x-2))/x

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                  /1    \
       tan(x)*atan|- - 2|
                  \x    /
f(x) = ------------------
               x
f(x)=tan⁡(x)atan⁡(−2+1x)xf{\left(x \right)} = \frac{\tan{\left(x \right)} \operatorname{atan}{\left(-2 + \frac{1}{x} \right)}}{x} 
f = (tan(x)*atan(-2 + 1/x))/x
The graph of the function
02468-8-6-4-2-1010-10050
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
tan⁡(x)atan⁡(−2+1x)x=0\frac{\tan{\left(x \right)} \operatorname{atan}{\left(-2 + \frac{1}{x} \right)}}{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=12x_{1} = \frac{1}{2}
x2=πx_{2} = \pi
Numerical solution
x1=72.2566310325652x_{1} = 72.2566310325652
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=3.14159265358979x_{3} = 3.14159265358979
x4=−43.9822971502571x_{4} = -43.9822971502571
x5=81.6814089933346x_{5} = 81.6814089933346
x6=−100.530964914873x_{6} = -100.530964914873
x7=28.2743338823081x_{7} = 28.2743338823081
x8=65.9734457253857x_{8} = 65.9734457253857
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=−9.42477796076938x_{10} = -9.42477796076938
x11=40.8407044966673x_{11} = 40.8407044966673
x12=56.5486677646163x_{12} = 56.5486677646163
x13=−56.5486677646163x_{13} = -56.5486677646163
x14=12.5663706143592x_{14} = 12.5663706143592
x15=43.9822971502571x_{15} = 43.9822971502571
x16=100.530964914873x_{16} = 100.530964914873
x17=−3.14159265358979x_{17} = -3.14159265358979
x18=−15.707963267949x_{18} = -15.707963267949
x19=59.6902604182061x_{19} = 59.6902604182061
x20=6.28318530717959x_{20} = 6.28318530717959
x21=9.42477796076938x_{21} = 9.42477796076938
x22=−53.4070751110265x_{22} = -53.4070751110265
x23=−47.1238898038469x_{23} = -47.1238898038469
x24=−87.9645943005142x_{24} = -87.9645943005142
x25=69.1150383789755x_{25} = 69.1150383789755
x26=21.9911485751286x_{26} = 21.9911485751286
x27=87.9645943005142x_{27} = 87.9645943005142
x28=18.8495559215388x_{28} = 18.8495559215388
x29=−84.8230016469244x_{29} = -84.8230016469244
x30=−72.2566310325652x_{30} = -72.2566310325652
x31=25.1327412287183x_{31} = 25.1327412287183
x32=37.6991118430775x_{32} = 37.6991118430775
x33=−25.1327412287183x_{33} = -25.1327412287183
x34=50.2654824574367x_{34} = 50.2654824574367
x35=34.5575191894877x_{35} = 34.5575191894877
x36=−6.28318530717959x_{36} = -6.28318530717959
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=−21.9911485751286x_{38} = -21.9911485751286
x39=−62.8318530717959x_{39} = -62.8318530717959
x40=75.398223686155x_{40} = 75.398223686155
x41=84.8230016469244x_{41} = 84.8230016469244
x42=53.4070751110265x_{42} = 53.4070751110265
x43=15.707963267949x_{43} = 15.707963267949
x44=−28.2743338823081x_{44} = -28.2743338823081
x45=−91.106186954104x_{45} = -91.106186954104
x46=47.1238898038469x_{46} = 47.1238898038469
x47=97.3893722612836x_{47} = 97.3893722612836
x48=−69.1150383789755x_{48} = -69.1150383789755
x49=94.2477796076938x_{49} = 94.2477796076938
x50=−18.8495559215388x_{50} = -18.8495559215388
x51=−50.2654824574367x_{51} = -50.2654824574367
x52=−37.6991118430775x_{52} = -37.6991118430775
x53=−81.6814089933346x_{53} = -81.6814089933346
x54=62.8318530717959x_{54} = 62.8318530717959
x55=78.5398163397448x_{55} = 78.5398163397448
x56=31.4159265358979x_{56} = 31.4159265358979
x57=−78.5398163397448x_{57} = -78.5398163397448
x58=−40.8407044966673x_{58} = -40.8407044966673
x59=−97.3893722612836x_{59} = -97.3893722612836
x60=−75.398223686155x_{60} = -75.398223686155
x61=91.106186954104x_{61} = 91.106186954104
x62=−12.5663706143592x_{62} = -12.5663706143592
x63=−94.2477796076938x_{63} = -94.2477796076938
x64=−34.5575191894877x_{64} = -34.5575191894877
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (tan(x)*atan(1/x - 2))/x.
tan⁡(0)atan⁡(−2+10)0\frac{\tan{\left(0 \right)} \operatorname{atan}{\left(-2 + \frac{1}{0} \right)}}{0}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
(tan⁡2(x)+1)atan⁡(−2+1x)−tan⁡(x)x2((−2+1x)2+1)x−tan⁡(x)atan⁡(−2+1x)x2=0\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(-2 + \frac{1}{x} \right)} - \frac{\tan{\left(x \right)}}{x^{2} \left(\left(-2 + \frac{1}{x}\right)^{2} + 1\right)}}{x} - \frac{\tan{\left(x \right)} \operatorname{atan}{\left(-2 + \frac{1}{x} \right)}}{x^{2}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Vertical asymptotes
Have:
x1=0x_{1} = 0
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→−∞(tan⁡(x)atan⁡(−2+1x)x)y = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \operatorname{atan}{\left(-2 + \frac{1}{x} \right)}}{x}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(tan⁡(x)atan⁡(−2+1x)x)y = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} \operatorname{atan}{\left(-2 + \frac{1}{x} \right)}}{x}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (tan(x)*atan(1/x - 2))/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→−∞(tan⁡(x)atan⁡(−2+1x)x2)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} \operatorname{atan}{\left(-2 + \frac{1}{x} \right)}}{x^{2}}\right)
True

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