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  • Graphing y =:
  • x+|x|
  • x/(x^2-4x+3)
  • x/((x-1)*(x-4))
  • x(x-1)^3

  • Identical expressions

  • -tan(one +atan(t^ two))
  • minus tangent of (1 plus arc tangent of gent of (t squared ))
  • minus tangent of (one plus arc tangent of gent of (t to the power of two))
  • -tan(1+atan(t2))
  • -tan1+atant2
  • -tan(1+atan(t²))
  • -tan(1+atan(t to the power of 2))
  • -tan1+atant^2

  • Similar expressions

  • tan(1+atan(t^2))
  • -tan(1-atan(t^2))
  • -tan(1+arctan(t^2))
Plotting a function graph/ -tan(1+atan(t^2))

Graphing y = -tan(1+atan(t^2))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           /        / 2\\
f(t) = -tan\1 + atan\t //
f(t)=−tan⁡(atan⁡(t2)+1)f{\left(t \right)} = - \tan{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)} 
f = -tan(atan(t^2) + 1)
The graph of the function
02468-8-6-4-2-1010-5050
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis T at f = 0
so we need to solve the equation:
−tan⁡(atan⁡(t2)+1)=0- \tan{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis T
The points of intersection with the Y axis coordinate
The graph crosses Y axis when t equals 0:
substitute t = 0 to -tan(1 + atan(t^2)).
−tan⁡(atan⁡(02)+1)- \tan{\left(\operatorname{atan}{\left(0^{2} \right)} + 1 \right)}
The result:
f(0)=−tan⁡(1)f{\left(0 \right)} = - \tan{\left(1 \right)}
The point:
(0, -tan(1))
Extrema of the function
In order to find the extrema, we need to solve the equation
ddtf(t)=0\frac{d}{d t} f{\left(t \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddtf(t)=\frac{d}{d t} f{\left(t \right)} =
the first derivative
−2t(tan⁡2(atan⁡(t2)+1)+1)t4+1=0- \frac{2 t \left(\tan^{2}{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)} + 1\right)}{t^{4} + 1} = 0
Solve this equation
The roots of this equation
t1=0t_{1} = 0
The values of the extrema at the points:
(0, -tan(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:
The function has no minima
Maxima of the function at points:
t1=0t_{1} = 0
Decreasing at intervals
(−∞,0]\left(-\infty, 0\right]
Increasing at intervals
[0,∞)\left[0, \infty\right)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dt2f(t)=0\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dt2f(t)=\frac{d^{2}}{d t^{2}} f{\left(t \right)} =
the second derivative
−2(tan⁡2(atan⁡(t2)+1)+1)(−4t4t4+1+4t2tan⁡(atan⁡(t2)+1)t4+1+1)t4+1=0- \frac{2 \left(\tan^{2}{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)} + 1\right) \left(- \frac{4 t^{4}}{t^{4} + 1} + \frac{4 t^{2} \tan{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)}}{t^{4} + 1} + 1\right)}{t^{4} + 1} = 0
Solve this equation
The roots of this equation
t1=−4561.0816054384t_{1} = -4561.0816054384
t2=6557.94206499265t_{2} = 6557.94206499265
t3=2194.45527257084t_{3} = 2194.45527257084
t4=−6960.37392835043t_{4} = -6960.37392835043
t5=−7178.47419056566t_{5} = -7178.47419056566
t6=4812.99995884256t_{6} = 4812.99995884256
t7=−4124.78172456999t_{7} = -4124.78172456999
t8=3285.84467120246t_{8} = 3285.84467120246
t9=−1942.21106929836t_{9} = -1942.21106929836
t10=5685.50017306841t_{10} = 5685.50017306841
t11=−2815.58612958771t_{11} = -2815.58612958771
t12=3940.40271652102t_{12} = 3940.40271652102
t13=10919.7658075958t_{13} = 10919.7658075958
t14=2849.38413704373t_{14} = 2849.38413704373
t15=−4342.93566505384t_{15} = -4342.93566505384
t16=−3033.8317153721t_{16} = -3033.8317153721
t17=3722.23025781143t_{17} = 3722.23025781143
t18=−2597.31114538297t_{18} = -2597.31114538297
t19=3504.04508557147t_{19} = 3504.04508557147
t20=−5433.60462135731t_{20} = -5433.60462135731
t21=9611.25862164469t_{21} = 9611.25862164469
t22=10047.4301664634t_{22} = 10047.4301664634
t23=7430.34617003321t_{23} = 7430.34617003321
t24=8956.99562917743t_{24} = 8956.99562917743
t25=9175.08412196212t_{25} = 9175.08412196212
t26=−10449.8277302572t_{26} = -10449.8277302572
t27=6339.83574361643t_{27} = 6339.83574361643
t28=−10885.9945779116t_{28} = -10885.9945779116
t29=6776.04606388126t_{29} = 6776.04606388126
t30=−1723.6948768428t_{30} = -1723.6948768428
t31=−2378.99869523626t_{31} = -2378.99869523626
t32=8738.90622506547t_{32} = 8738.90622506547
t33=−10667.9114237547t_{33} = -10667.9114237547
t34=−3470.25683552187t_{34} = -3470.25683552187
t35=−6524.16738262466t_{35} = -6524.16738262466
t36=8084.63180744039t_{36} = 8084.63180744039
t37=−8487.04335987365t_{37} = -8487.04335987365
t38=−9359.39985337781t_{38} = -9359.39985337781
t39=8302.72439567443t_{39} = 8302.72439567443
t40=−6742.27172393545t_{40} = -6742.27172393545
t41=7866.53797982263t_{41} = 7866.53797982263
t42=−3688.44417247294t_{42} = -3688.44417247294
t43=5467.38167722636t_{43} = 5467.38167722636
t44=6121.72685162975t_{44} = 6121.72685162975
t45=7212.24793668016t_{45} = 7212.24793668016
t46=−10231.7434629383t_{46} = -10231.7434629383
t47=−7832.76494582429t_{47} = -7832.76494582429
t48=4158.56446033742t_{48} = 4158.56446033742
t49=−4779.22064019992t_{49} = -4779.22064019992
t50=5249.25917265879t_{50} = 5249.25917265879
t51=−7396.57268232278t_{51} = -7396.57268232278
t52=10701.6827335228t_{52} = 10701.6827335228
t53=−6306.06068280019t_{53} = -6306.06068280019
t54=5031.13213833407t_{54} = 5031.13213833407
t55=−8268.95174565771t_{55} = -8268.95174565771
t56=2631.11413630212t_{56} = 2631.11413630212
t57=−9795.57305356333t_{57} = -9795.57305356333
t58=4376.71708985591t_{58} = 4376.71708985591
t59=1757.53994666307t_{59} = 1757.53994666307
t60=6994.14795746859t_{60} = 6994.14795746859
t61=−9577.48682614751t_{61} = -9577.48682614751
t62=7648.44280683045t_{62} = 7648.44280683045
t63=9829.34473883156t_{63} = 9829.34473883156
t64=−7614.66955569501t_{64} = -7614.66955569501
t65=−2160.63746348536t_{65} = -2160.63746348536
t66=2412.80808854532t_{66} = 2412.80808854532
t67=9393.17176688301t_{67} = 9393.17176688301
t68=10483.5991251663t_{68} = 10483.5991251663
t69=1976.04024923119t_{69} = 1976.04024923119
t70=4594.86190201511t_{70} = 4594.86190201511
t71=−3252.05380941148t_{71} = -3252.05380941148
t72=10265.5149484802t_{72} = 10265.5149484802
t73=−8923.22345322829t_{73} = -8923.22345322829
t74=−3906.61844534581t_{74} = -3906.61844534581
t75=8520.81583966705t_{75} = 8520.81583966705
t76=−10013.6585843141t_{76} = -10013.6585843141
t77=−8705.13390288973t_{77} = -8705.13390288973
t78=−5651.72370321664t_{78} = -5651.72370321664
t79=−4997.3536728881t_{79} = -4997.3536728881
t80=3067.62576804298t_{80} = 3067.62576804298
t81=−8050.8589732138t_{81} = -8050.8589732138
t82=−5869.83915653713t_{82} = -5869.83915653713
t83=−9141.31208192029t_{83} = -9141.31208192029
t84=−6087.95137110257t_{84} = -6087.95137110257
t85=5903.61510426179t_{85} = 5903.61510426179
t86=−5215.48145601722t_{86} = -5215.48145601722

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:
Have no bends at the whole real axis
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at t->+oo and t->-oo
lim⁡t→−∞(−tan⁡(atan⁡(t2)+1))=⟨−∞,∞⟩\lim_{t \to -\infty}\left(- \tan{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡t→∞(−tan⁡(atan⁡(t2)+1))=⟨−∞,∞⟩\lim_{t \to \infty}\left(- \tan{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-t) и f = -f(-t).
So, check:
−tan⁡(atan⁡(t2)+1)=−tan⁡(atan⁡(t2)+1)- \tan{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)} = - \tan{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)}
- Yes
−tan⁡(atan⁡(t2)+1)=tan⁡(atan⁡(t2)+1)- \tan{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)} = \tan{\left(\operatorname{atan}{\left(t^{2} \right)} + 1 \right)}
- No
so, the function
is
even
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