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sinh(tanh(x))
Plotting a function graph/ sinh(tanh(x))

Graphing y = sinh(tanh(x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = sinh(tanh(x))
f(x)=sinh⁡(tanh⁡(x))f{\left(x \right)} = \sinh{\left(\tanh{\left(x \right)} \right)} 
f = sinh(tanh(x))
The graph of the function
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.02.5-2.5
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:
sinh⁡(tanh⁡(x))=0\sinh{\left(\tanh{\left(x \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sinh(tanh(x)).
sinh⁡(tanh⁡(0))\sinh{\left(\tanh{\left(0 \right)} \right)}
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−tanh⁡2(x))cosh⁡(tanh⁡(x))=0\left(1 - \tanh^{2}{\left(x \right)}\right) \cosh{\left(\tanh{\left(x \right)} \right)} = 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
((tanh⁡2(x)−1)sinh⁡(tanh⁡(x))+2cosh⁡(tanh⁡(x))tanh⁡(x))(tanh⁡2(x)−1)=0\left(\left(\tanh^{2}{\left(x \right)} - 1\right) \sinh{\left(\tanh{\left(x \right)} \right)} + 2 \cosh{\left(\tanh{\left(x \right)} \right)} \tanh{\left(x \right)}\right) \left(\tanh^{2}{\left(x \right)} - 1\right) = 0
Solve this equation
The roots of this equation
x1=−22.36613259083x_{1} = -22.36613259083
x2=−72x_{2} = -72
x3=−62x_{3} = -62
x4=−44x_{4} = -44
x5=−54x_{5} = -54
x6=31.8870724794909x_{6} = 31.8870724794909
x7=−98x_{7} = -98
x8=96x_{8} = 96
x9=64x_{9} = 64
x10=28.6760419286256x_{10} = 28.6760419286256
x11=20.6753972494872x_{11} = 20.6753972494872
x12=−31.976245996576x_{12} = -31.976245996576
x13=54x_{13} = 54
x14=62x_{14} = 62
x15=40x_{15} = 40
x16=−48x_{16} = -48
x17=−96x_{17} = -96
x18=0x_{18} = 0
x19=−40x_{19} = -40
x20=18.6753976927522x_{20} = 18.6753976927522
x21=100x_{21} = 100
x22=78x_{22} = 78
x23=−30.3746613927776x_{23} = -30.3746613927776
x24=−32x_{24} = -32
x25=80x_{25} = 80
x26=−20.3661326061554x_{26} = -20.3661326061554
x27=−52x_{27} = -52
x28=32.468253968254x_{28} = 32.468253968254
x29=70x_{29} = 70
x30=16.6754218978551x_{30} = 16.6754218978551
x31=−100x_{31} = -100
x32=−16.3661781103476x_{32} = -16.3661781103476
x33=32x_{33} = 32
x34=44x_{34} = 44
x35=60x_{35} = 60
x36=90x_{36} = 90
x37=−32.2038153090912x_{37} = -32.2038153090912
x38=76x_{38} = 76
x39=72x_{39} = 72
x40=86x_{40} = 86
x41=56x_{41} = 56
x42=−58x_{42} = -58
x43=−31.7529322617035x_{43} = -31.7529322617035
x44=32.000015780841x_{44} = 32.000015780841
x45=−86x_{45} = -86
x46=−56x_{46} = -56
x47=42x_{47} = 42
x48=−26.3661315069563x_{48} = -26.3661315069563
x49=31.7312936927894x_{49} = 31.7312936927894
x50=98x_{50} = 98
x51=22.6753972443119x_{51} = 22.6753972443119
x52=−80x_{52} = -80
x53=−76x_{53} = -76
x54=84x_{54} = 84
x55=−84x_{55} = -84
x56=94x_{56} = 94
x57=58x_{57} = 58
x58=31.7787220654169x_{58} = 31.7787220654169
x59=−38x_{59} = -38
x60=34x_{60} = 34
x61=30.6502781482389x_{61} = 30.6502781482389
x62=−18.3661334245818x_{62} = -18.3661334245818
x63=−82x_{63} = -82
x64=−92x_{64} = -92
x65=38x_{65} = 38
x66=−42x_{66} = -42
x67=88x_{67} = 88
x68=−78x_{68} = -78
x69=−36x_{69} = -36
x70=−60x_{70} = -60
x71=−28.3660709532537x_{71} = -28.3660709532537
x72=92x_{72} = 92
x73=−64x_{73} = -64
x74=−88x_{74} = -88
x75=24.6753973204929x_{75} = 24.6753973204929
x76=−24.3661326142577x_{76} = -24.3661326142577
x77=−66x_{77} = -66
x78=36x_{78} = 36
x79=−46x_{79} = -46
x80=26.6753908017134x_{80} = 26.6753908017134
x81=66x_{81} = 66
x82=−34x_{82} = -34
x83=52x_{83} = 52
x84=−94x_{84} = -94
x85=−90x_{85} = -90
x86=68x_{86} = 68
x87=82x_{87} = 82
x88=74x_{88} = 74
x89=−68x_{89} = -68
x90=−74x_{90} = -74
x91=46x_{91} = 46
x92=50x_{92} = 50
x93=−31.3350186083881x_{93} = -31.3350186083881
x94=−70x_{94} = -70
x95=−50x_{95} = -50
x96=48x_{96} = 48
x97=−31.5673076923077x_{97} = -31.5673076923077

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
(−∞,0]\left(-\infty, 0\right]
Convex at the intervals
[0,∞)\left[0, \infty\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:
sinh⁡(tanh⁡(x))=−sinh⁡(tanh⁡(x))\sinh{\left(\tanh{\left(x \right)} \right)} = - \sinh{\left(\tanh{\left(x \right)} \right)}
- No
sinh⁡(tanh⁡(x))=sinh⁡(tanh⁡(x))\sinh{\left(\tanh{\left(x \right)} \right)} = \sinh{\left(\tanh{\left(x \right)} \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = sinh(tanh(x))
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