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Plotting a function graph/ log(cos(2*x-3))

Graphing y = log(cos(2*x-3))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = log(cos(2*x - 3))
f(x)=log⁡(cos⁡(2x−3))f{\left(x \right)} = \log{\left(\cos{\left(2 x - 3 \right)} \right)} 
f = log(cos(2*x - 3))
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⁡(cos⁡(2x−3))=0\log{\left(\cos{\left(2 x - 3 \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=32x_{1} = \frac{3}{2}
x2=32+πx_{2} = \frac{3}{2} + \pi
Numerical solution
x1=7.78318547397097x_{1} = 7.78318547397097
x2=92.6061870252117x_{2} = 92.6061870252117
x3=−83.3230019549508x_{3} = -83.3230019549508
x4=−42.4822967419943x_{4} = -42.4822967419943
x5=20.3495558571988x_{5} = 20.3495558571988
x6=−61.3318533698126x_{6} = -61.3318533698126
x7=−77.0398164814975x_{7} = -77.0398164814975
x8=−80.1814088517395x_{8} = -80.1814088517395
x9=−17.3495562007102x_{9} = -17.3495562007102
x10=−45.6238898840372x_{10} = -45.6238898840372
x11=−20.4911481873041x_{11} = -20.4911481873041
x12=70.6150381336808x_{12} = 70.6150381336808
x13=−42.482296664075x_{13} = -42.482296664075
x14=1.50000053642386x_{14} = 1.50000053642386
x15=89.4645945782532x_{15} = 89.4645945782532
x16=−86.4645938548753x_{16} = -86.4645938548753
x17=−95.8893718226417x_{17} = -95.8893718226417
x18=−1.6415927232245x_{18} = -1.6415927232245
x19=45.4822977551994x_{19} = 45.4822977551994
x20=−64.4734452604177x_{20} = -64.4734452604177
x21=83.1814089792103x_{21} = 83.1814089792103
x22=42.3407044376452x_{22} = 42.3407044376452
x23=−17.349555936709x_{23} = -17.349555936709
x24=4.64159238332314x_{24} = 4.64159238332314
x25=−89.6061870448566x_{25} = -89.6061870448566
x26=14.0663704209511x_{26} = 14.0663704209511
x27=23.4911489192456x_{27} = 23.4911489192456
x28=48.6238895504951x_{28} = 48.6238895504951
x29=−7.92477827673125x_{29} = -7.92477827673125
x30=−26.7743338803417x_{30} = -26.7743338803417
x31=−70.7566310225095x_{31} = -70.7566310225095
x32=73.7566312191606x_{32} = 73.7566312191606
x33=−36.1991116890236x_{33} = -36.1991116890236
x34=26.63274096705x_{34} = 26.63274096705
x35=−14.2079631075661x_{35} = -14.2079631075661
x36=45.4822974719891x_{36} = 45.4822974719891
x37=98.8893714973164x_{37} = 98.8893714973164
x38=−99.0309650408064x_{38} = -99.0309650408064
x39=64.3318530181245x_{39} = 64.3318530181245
x40=70.6150384310865x_{40} = 70.6150384310865
x41=26.6327412439176x_{41} = 26.6327412439176
x42=−48.7654824515488x_{42} = -48.7654824515488
x43=−55.0486679240968x_{43} = -55.0486679240968
x44=−23.6327413036338x_{44} = -23.6327413036338
x45=4.64159265218678x_{45} = 4.64159265218678
x46=17.2079632627963x_{46} = 17.2079632627963
x47=−64.4734452982655x_{47} = -64.4734452982655
x48=−86.4645938566005x_{48} = -86.4645938566005
x49=95.7477798011181x_{49} = 95.7477798011181
x50=−67.6150384644424x_{50} = -67.6150384644424
x51=−92.747779593219x_{51} = -92.747779593219
x52=80.039816271268x_{52} = 80.039816271268
x53=48.6238898372022x_{53} = 48.6238898372022
x54=36.0575190562039x_{54} = 36.0575190562039
x55=29.7743340555976x_{55} = 29.7743340555976
x56=−39.3407047850781x_{56} = -39.3407047850781
x57=−20.4911480653688x_{57} = -20.4911480653688
x58=86.3230015986475x_{58} = 86.3230015986475
x59=51.765482637324x_{59} = 51.765482637324
x60=61.1902604068607x_{60} = 61.1902604068607
x61=58.0486676692494x_{61} = 58.0486676692494
x62=67.4734463806069x_{62} = 67.4734463806069
x63=−4.78318530889195x_{63} = -4.78318530889195
x64=23.4911491425042x_{64} = 23.4911491425042
x65=39.1991118347172x_{65} = 39.1991118347172
x66=92.6061867166268x_{66} = 92.6061867166268
x67=−58.1902602704114x_{67} = -58.1902602704114
x68=−20.4911482270984x_{68} = -20.4911482270984
x69=1.50000036628856x_{69} = 1.50000036628856
x70=67.4734460248885x_{70} = 67.4734460248885
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(cos(2*x - 3)).
log⁡(cos⁡(−3+0⋅2))\log{\left(\cos{\left(-3 + 0 \cdot 2 \right)} \right)}
The result:
f(0)=log⁡(−cos⁡(3))+iπf{\left(0 \right)} = \log{\left(- \cos{\left(3 \right)} \right)} + i \pi
The point:
(0, pi*i + log(-cos(3)))
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
−2sin⁡(2x−3)cos⁡(2x−3)=0- \frac{2 \sin{\left(2 x - 3 \right)}}{\cos{\left(2 x - 3 \right)}} = 0
Solve this equation
The roots of this equation
x1=32x_{1} = \frac{3}{2}
x2=32+π2x_{2} = \frac{3}{2} + \frac{\pi}{2}
The values of the extrema at the points:
(3/2, 0)

 3   pi       
(- + --, pi*I)
 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=32x_{2} = \frac{3}{2}
Decreasing at intervals
(−∞,32]\left(-\infty, \frac{3}{2}\right]
Increasing at intervals
[32,∞)\left[\frac{3}{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
−4(sin⁡2(2x−3)cos⁡2(2x−3)+1)=0- 4 \left(\frac{\sin^{2}{\left(2 x - 3 \right)}}{\cos^{2}{\left(2 x - 3 \right)}} + 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⁡(cos⁡(2x−3))=log⁡(⟨−1,1⟩)\lim_{x \to -\infty} \log{\left(\cos{\left(2 x - 3 \right)} \right)} = \log{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=log⁡(⟨−1,1⟩)y = \log{\left(\left\langle -1, 1\right\rangle \right)}
lim⁡x→∞log⁡(cos⁡(2x−3))=log⁡(⟨−1,1⟩)\lim_{x \to \infty} \log{\left(\cos{\left(2 x - 3 \right)} \right)} = \log{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=log⁡(⟨−1,1⟩)y = \log{\left(\left\langle -1, 1\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(cos(2*x - 3)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(cos⁡(2x−3))x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(2 x - 3 \right)} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(cos⁡(2x−3))x)=0\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(2 x - 3 \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⁡(cos⁡(2x−3))=log⁡(cos⁡(2x+3))\log{\left(\cos{\left(2 x - 3 \right)} \right)} = \log{\left(\cos{\left(2 x + 3 \right)} \right)}
- No
log⁡(cos⁡(2x−3))=−log⁡(cos⁡(2x+3))\log{\left(\cos{\left(2 x - 3 \right)} \right)} = - \log{\left(\cos{\left(2 x + 3 \right)} \right)}
- No
so, the function
not is
neither even, nor odd
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