Mister Exam
Graphing y = y=|tg(2x+pi/3)|
The solution
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| / pi\|
f(x) = |tan|2*x + --||
| \ 3 /|
$$f{\left(x \right)} = \left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right|$$
f = Abs(tan(2*x + pi/3))
The graph of the function
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:
$$\left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{6}$$
Numerical solution
$$x_{1} = 84.2994028713261$$
$$x_{2} = 32.4631240870945$$
$$x_{3} = 30.8923277602996$$
$$x_{4} = -44.5058959258554$$
$$x_{5} = 65.4498469497874$$
$$x_{6} = -8.37758040957278$$
$$x_{7} = 90.5825881785057$$
$$x_{8} = -96.342174710087$$
$$x_{9} = 40.317105721069$$
$$x_{10} = 82.7286065445312$$
$$x_{11} = 38.7463093942741$$
$$x_{12} = 96.8657734856853$$
$$x_{13} = 35.6047167406843$$
$$x_{14} = -46.0766922526503$$
$$x_{15} = 63.8790506229925$$
$$x_{16} = -49.2182849062401$$
$$x_{17} = 41.8879020478639$$
$$x_{18} = -82.2050077689329$$
$$x_{19} = 48.1710873550435$$
$$x_{20} = 87.4409955249159$$
$$x_{21} = 13.6135681655558$$
$$x_{22} = 43.4586983746588$$
$$x_{23} = 56.025068989018$$
$$x_{24} = 74.8746249105567$$
$$x_{25} = -75.9218224617533$$
$$x_{26} = -3.66519142918809$$
$$x_{27} = 5.75958653158129$$
$$x_{28} = -19.3731546971371$$
$$x_{29} = 85.870199198121$$
$$x_{30} = 7.33038285837618$$
$$x_{31} = -16.2315620435473$$
$$x_{32} = 4.18879020478639$$
$$x_{33} = -5.23598775598299$$
$$x_{34} = -74.3510261349584$$
$$x_{35} = -17.8023583703422$$
$$x_{36} = 34.0339204138894$$
$$x_{37} = -2.0943951023932$$
$$x_{38} = 52.8834763354282$$
$$x_{39} = 12.0427718387609$$
$$x_{40} = 49.7418836818384$$
$$x_{41} = 98.4365698124802$$
$$x_{42} = -0.523598775598299$$
$$x_{43} = 27.7507351067098$$
$$x_{44} = 79.5870138909414$$
$$x_{45} = 16.7551608191456$$
$$x_{46} = -24.0855436775217$$
$$x_{47} = 54.4542726622231$$
$$x_{48} = 26.1799387799149$$
$$x_{49} = 2.61799387799149$$
$$x_{50} = 24.60914245312$$
$$x_{51} = -41.3643032722656$$
$$x_{52} = 10.471975511966$$
$$x_{53} = -9.94837673636768$$
$$x_{54} = 19.8967534727354$$
$$x_{55} = -69.6386371545737$$
$$x_{56} = 78.0162175641465$$
$$x_{57} = 57.5958653158129$$
$$x_{58} = -90.0589894029074$$
$$x_{59} = 60.7374579694027$$
$$x_{60} = 18.3259571459405$$
$$x_{61} = -55.5014702134197$$
$$x_{62} = -52.3598775598299$$
$$x_{63} = -99.4837673636768$$
$$x_{64} = -88.4881930761125$$
$$x_{65} = -66.497044500984$$
$$x_{66} = 717.330322569669$$
$$x_{67} = -60.2138591938044$$
$$x_{68} = -77.4926187885482$$
$$x_{69} = 46.6002910282486$$
$$x_{70} = 68.5914396033772$$
$$x_{71} = -61.7846555205993$$
$$x_{72} = 21.4675497995303$$
$$x_{73} = -25.6563400043166$$
$$x_{74} = -58.6430628670095$$
$$x_{75} = 93.7241808320955$$
$$x_{76} = -68.0678408277789$$
$$x_{77} = 100.007366139275$$
$$x_{78} = -91.6297857297023$$
$$x_{79} = -47.6474885794452$$
$$x_{80} = -33.5103216382911$$
$$x_{81} = -30.3687289847013$$
$$x_{82} = -93.2005820564972$$
$$x_{83} = -11.5191730631626$$
$$x_{84} = 76.4454212373516$$
$$x_{85} = -80.634211442138$$
$$x_{86} = -38.2227106186758$$
$$x_{87} = -71.2094334813686$$
$$x_{88} = -63.3554518473942$$
$$x_{89} = -83.7758040957278$$
$$x_{90} = -27.2271363311115$$
$$x_{91} = 71.733032256967$$
$$x_{92} = 92.1533845053006$$
$$x_{93} = -39.7935069454707$$
$$x_{94} = 62.3082542961976$$
$$x_{95} = -85.3466004225227$$
$$x_{96} = 70.162235930172$$
$$x_{97} = -22.5147473507269$$
so we need to solve the equation:
$$\left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{6}$$
Numerical solution
$$x_{1} = 84.2994028713261$$
$$x_{2} = 32.4631240870945$$
$$x_{3} = 30.8923277602996$$
$$x_{4} = -44.5058959258554$$
$$x_{5} = 65.4498469497874$$
$$x_{6} = -8.37758040957278$$
$$x_{7} = 90.5825881785057$$
$$x_{8} = -96.342174710087$$
$$x_{9} = 40.317105721069$$
$$x_{10} = 82.7286065445312$$
$$x_{11} = 38.7463093942741$$
$$x_{12} = 96.8657734856853$$
$$x_{13} = 35.6047167406843$$
$$x_{14} = -46.0766922526503$$
$$x_{15} = 63.8790506229925$$
$$x_{16} = -49.2182849062401$$
$$x_{17} = 41.8879020478639$$
$$x_{18} = -82.2050077689329$$
$$x_{19} = 48.1710873550435$$
$$x_{20} = 87.4409955249159$$
$$x_{21} = 13.6135681655558$$
$$x_{22} = 43.4586983746588$$
$$x_{23} = 56.025068989018$$
$$x_{24} = 74.8746249105567$$
$$x_{25} = -75.9218224617533$$
$$x_{26} = -3.66519142918809$$
$$x_{27} = 5.75958653158129$$
$$x_{28} = -19.3731546971371$$
$$x_{29} = 85.870199198121$$
$$x_{30} = 7.33038285837618$$
$$x_{31} = -16.2315620435473$$
$$x_{32} = 4.18879020478639$$
$$x_{33} = -5.23598775598299$$
$$x_{34} = -74.3510261349584$$
$$x_{35} = -17.8023583703422$$
$$x_{36} = 34.0339204138894$$
$$x_{37} = -2.0943951023932$$
$$x_{38} = 52.8834763354282$$
$$x_{39} = 12.0427718387609$$
$$x_{40} = 49.7418836818384$$
$$x_{41} = 98.4365698124802$$
$$x_{42} = -0.523598775598299$$
$$x_{43} = 27.7507351067098$$
$$x_{44} = 79.5870138909414$$
$$x_{45} = 16.7551608191456$$
$$x_{46} = -24.0855436775217$$
$$x_{47} = 54.4542726622231$$
$$x_{48} = 26.1799387799149$$
$$x_{49} = 2.61799387799149$$
$$x_{50} = 24.60914245312$$
$$x_{51} = -41.3643032722656$$
$$x_{52} = 10.471975511966$$
$$x_{53} = -9.94837673636768$$
$$x_{54} = 19.8967534727354$$
$$x_{55} = -69.6386371545737$$
$$x_{56} = 78.0162175641465$$
$$x_{57} = 57.5958653158129$$
$$x_{58} = -90.0589894029074$$
$$x_{59} = 60.7374579694027$$
$$x_{60} = 18.3259571459405$$
$$x_{61} = -55.5014702134197$$
$$x_{62} = -52.3598775598299$$
$$x_{63} = -99.4837673636768$$
$$x_{64} = -88.4881930761125$$
$$x_{65} = -66.497044500984$$
$$x_{66} = 717.330322569669$$
$$x_{67} = -60.2138591938044$$
$$x_{68} = -77.4926187885482$$
$$x_{69} = 46.6002910282486$$
$$x_{70} = 68.5914396033772$$
$$x_{71} = -61.7846555205993$$
$$x_{72} = 21.4675497995303$$
$$x_{73} = -25.6563400043166$$
$$x_{74} = -58.6430628670095$$
$$x_{75} = 93.7241808320955$$
$$x_{76} = -68.0678408277789$$
$$x_{77} = 100.007366139275$$
$$x_{78} = -91.6297857297023$$
$$x_{79} = -47.6474885794452$$
$$x_{80} = -33.5103216382911$$
$$x_{81} = -30.3687289847013$$
$$x_{82} = -93.2005820564972$$
$$x_{83} = -11.5191730631626$$
$$x_{84} = 76.4454212373516$$
$$x_{85} = -80.634211442138$$
$$x_{86} = -38.2227106186758$$
$$x_{87} = -71.2094334813686$$
$$x_{88} = -63.3554518473942$$
$$x_{89} = -83.7758040957278$$
$$x_{90} = -27.2271363311115$$
$$x_{91} = 71.733032256967$$
$$x_{92} = 92.1533845053006$$
$$x_{93} = -39.7935069454707$$
$$x_{94} = 62.3082542961976$$
$$x_{95} = -85.3466004225227$$
$$x_{96} = 70.162235930172$$
$$x_{97} = -22.5147473507269$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\left(2 \tan^{2}{\left(2 x + \frac{\pi}{3} \right)} + 2\right) \operatorname{sign}{\left(\tan{\left(2 x + \frac{\pi}{3} \right)} \right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\left(2 \tan^{2}{\left(2 x + \frac{\pi}{3} \right)} + 2\right) \operatorname{sign}{\left(\tan{\left(2 x + \frac{\pi}{3} \right)} \right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right|$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right|$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right|$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right|$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(tan(2*x + pi/3)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right|}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right|}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right|}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right|}{x}\right)$$
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:
$$\left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right| = \sqrt{- \tan{\left(2 x - \frac{\pi}{3} \right)} \cot{\left(2 x + \frac{\pi}{6} \right)}}$$
- No
$$\left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right| = - \sqrt{- \tan{\left(2 x - \frac{\pi}{3} \right)} \cot{\left(2 x + \frac{\pi}{6} \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right| = \sqrt{- \tan{\left(2 x - \frac{\pi}{3} \right)} \cot{\left(2 x + \frac{\pi}{6} \right)}}$$
- No
$$\left|{\tan{\left(2 x + \frac{\pi}{3} \right)}}\right| = - \sqrt{- \tan{\left(2 x - \frac{\pi}{3} \right)} \cot{\left(2 x + \frac{\pi}{6} \right)}}$$
- No
so, the function
not is
neither even, nor odd