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Plotting a function graph/ ln(sqrt(2)sin(x+1))

Graphing y = ln(sqrt(2)sin(x+1))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = log\\/ 2 *sin(x + 1)/
f(x)=log⁡(2sin⁡(x+1))f{\left(x \right)} = \log{\left(\sqrt{2} \sin{\left(x + 1 \right)} \right)} 
f = log(sqrt(2)*sin(x + 1))
The graph of the function
02468-8-6-4-2-10105-5
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:
log⁡(2sin⁡(x+1))=0\log{\left(\sqrt{2} \sin{\left(x + 1 \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−1+π4x_{1} = -1 + \frac{\pi}{4}
x2=−1+3π4x_{2} = -1 + \frac{3 \pi}{4}
Numerical solution
x1=1.35619449019234x_{1} = 1.35619449019234
x2=32.7721210260903x_{2} = 32.7721210260903
x3=−19.0641577581413x_{3} = -19.0641577581413
x4=43.7676953136546x_{4} = 43.7676953136546
x5=−94.4623814442964x_{5} = -94.4623814442964
x6=−30.0597320457056x_{6} = -30.0597320457056
x7=−42.6261026600648x_{7} = -42.6261026600648
x8=13.9225651045515x_{8} = 13.9225651045515
x9=−17.4933614313464x_{9} = -17.4933614313464
x10=68.9004365423729x_{10} = 68.9004365423729
x11=7.63937979737193x_{11} = 7.63937979737193
x12=94.0331777710912x_{12} = 94.0331777710912
x13=−6.49778714378214x_{13} = -6.49778714378214
x14=−99.174770424681x_{14} = -99.174770424681
x15=−67.7588438887831x_{15} = -67.7588438887831
x16=89.3207887907066x_{16} = 89.3207887907066
x17=57.9048622548086x_{17} = 57.9048622548086
x18=−0.214601836602552x_{18} = -0.214601836602552
x19=−48.9092879672443x_{19} = -48.9092879672443
x20=−11.2101761241668x_{20} = -11.2101761241668
x21=37.484510006475x_{21} = 37.484510006475
x22=−37.9137136796801x_{22} = -37.9137136796801
x23=−25.3473430653209x_{23} = -25.3473430653209
x24=62.6172512351933x_{24} = 62.6172512351933
x25=12.3517687777566x_{25} = 12.3517687777566
x26=56.3340659280137x_{26} = 56.3340659280137
x27=45.3384916404494x_{27} = 45.3384916404494
x28=−86.6083998103219x_{28} = -86.6083998103219
x29=−63.0464549083984x_{29} = -63.0464549083984
x30=−50.4800842940392x_{30} = -50.4800842940392
x31=−92.8915851175014x_{31} = -92.8915851175014
x32=87.7499924639117x_{32} = 87.7499924639117
x33=−23.776546738526x_{33} = -23.776546738526
x34=−74.0420291959627x_{34} = -74.0420291959627
x35=95.6039740978861x_{35} = 95.6039740978861
x36=70.4712328691678x_{36} = 70.4712328691678
x37=101.887159405066x_{37} = 101.887159405066
x38=−75.6128255227576x_{38} = -75.6128255227576
x39=−56.7632696012188x_{39} = -56.7632696012188
x40=26.4889357189107x_{40} = 26.4889357189107
x41=−61.4756585816035x_{41} = -61.4756585816035
x42=39.0553063332699x_{42} = 39.0553063332699
x43=24.9181393921158x_{43} = 24.9181393921158
x44=−69.329640215578x_{44} = -69.329640215578
x45=−44.1968989868597x_{45} = -44.1968989868597
x46=−4.92699081698724x_{46} = -4.92699081698724
x47=75.1836218495525x_{47} = 75.1836218495525
x48=76.7544181763474x_{48} = 76.7544181763474
x49=81.4668071567321x_{49} = 81.4668071567321
x50=−55.1924732744239x_{50} = -55.1924732744239
x51=−36.3429173528852x_{51} = -36.3429173528852
x52=18.6349540849362x_{52} = 18.6349540849362
x53=83.037603483527x_{53} = 83.037603483527
x54=−80.3252145031423x_{54} = -80.3252145031423
x55=64.1880475619882x_{55} = 64.1880475619882
x56=100.316363078271x_{56} = 100.316363078271
x57=−88.1791961371168x_{57} = -88.1791961371168
x58=−12.7809724509617x_{58} = -12.7809724509617
x59=−81.8960108299372x_{59} = -81.8960108299372
x60=31.2013246992954x_{60} = 31.2013246992954
x61=−31.6305283725005x_{61} = -31.6305283725005
x62=50.0508806208341x_{62} = 50.0508806208341
x63=20.2057504117311x_{63} = 20.2057504117311
x64=6.06858347057703x_{64} = 6.06858347057703
x65=51.621676947629x_{65} = 51.621676947629
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(sqrt(2)*sin(x + 1)).
log⁡(2sin⁡(1))\log{\left(\sqrt{2} \sin{\left(1 \right)} \right)}
The result:
f(0)=log⁡(2sin⁡(1))f{\left(0 \right)} = \log{\left(\sqrt{2} \sin{\left(1 \right)} \right)}
The point:
(0, log(sqrt(2)*sin(1)))
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
cos⁡(x+1)sin⁡(x+1)=0\frac{\cos{\left(x + 1 \right)}}{\sin{\left(x + 1 \right)}} = 0
Solve this equation
The roots of this equation
x1=−1+π2x_{1} = -1 + \frac{\pi}{2}
x2=−1+3π2x_{2} = -1 + \frac{3 \pi}{2}
The values of the extrema at the points:
      pi     /  ___\ 
(-1 + --, log\\/ 2 /)
      2              

      3*pi            /  ___\ 
(-1 + ----, pi*I + log\\/ 2 /)
       2


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:
x2=−1+π2x_{2} = -1 + \frac{\pi}{2}
Decreasing at intervals
(−∞,−1+π2]\left(-\infty, -1 + \frac{\pi}{2}\right]
Increasing at intervals
[−1+π2,∞)\left[-1 + \frac{\pi}{2}, \infty\right)
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
−(1+cos⁡2(x+1)sin⁡2(x+1))=0- (1 + \frac{\cos^{2}{\left(x + 1 \right)}}{\sin^{2}{\left(x + 1 \right)}}) = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞log⁡(2sin⁡(x+1))=log⁡(2⟨−1,1⟩)\lim_{x \to -\infty} \log{\left(\sqrt{2} \sin{\left(x + 1 \right)} \right)} = \log{\left(\sqrt{2} \left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=log⁡(2⟨−1,1⟩)y = \log{\left(\sqrt{2} \left\langle -1, 1\right\rangle \right)}
lim⁡x→∞log⁡(2sin⁡(x+1))=log⁡(2⟨−1,1⟩)\lim_{x \to \infty} \log{\left(\sqrt{2} \sin{\left(x + 1 \right)} \right)} = \log{\left(\sqrt{2} \left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=log⁡(2⟨−1,1⟩)y = \log{\left(\sqrt{2} \left\langle -1, 1\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(sqrt(2)*sin(x + 1)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(2sin⁡(x+1))x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\sqrt{2} \sin{\left(x + 1 \right)} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(2sin⁡(x+1))x)=0\lim_{x \to \infty}\left(\frac{\log{\left(\sqrt{2} \sin{\left(x + 1 \right)} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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⁡(2sin⁡(x+1))=log⁡(−2sin⁡(x−1))\log{\left(\sqrt{2} \sin{\left(x + 1 \right)} \right)} = \log{\left(- \sqrt{2} \sin{\left(x - 1 \right)} \right)}
- No
log⁡(2sin⁡(x+1))=−log⁡(−2sin⁡(x−1))\log{\left(\sqrt{2} \sin{\left(x + 1 \right)} \right)} = - \log{\left(- \sqrt{2} \sin{\left(x - 1 \right)} \right)}
- No
so, the function
not is
neither even, nor odd
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