Mister Exam
Graphing y = tg+|tgx|
The solution
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f(x) = tan(x) + |tan(x)|
$$f{\left(x \right)} = \tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|$$
f = tan(x) + Abs(tan(x))
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:
$$\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -70$$
$$x_{2} = 62$$
$$x_{3} = 90$$
$$x_{4} = 50$$
$$x_{5} = -16$$
$$x_{6} = 94$$
$$x_{7} = -76$$
$$x_{8} = -18.8495559215388$$
$$x_{9} = -36$$
$$x_{10} = 25.1327412287183$$
$$x_{11} = -58$$
$$x_{12} = 8$$
$$x_{13} = 81.6814089933346$$
$$x_{14} = 30$$
$$x_{15} = -22$$
$$x_{16} = 69.1150383789755$$
$$x_{17} = -26$$
$$x_{18} = -88$$
$$x_{19} = -78.5398163397448$$
$$x_{20} = -98$$
$$x_{21} = 36.25$$
$$x_{22} = 6$$
$$x_{23} = -10$$
$$x_{24} = -95.75$$
$$x_{25} = 53.4070751110265$$
$$x_{26} = 84$$
$$x_{27} = 12$$
$$x_{28} = -91.106186954104$$
$$x_{29} = 78$$
$$x_{30} = 80.25$$
$$x_{31} = -72.2566310325652$$
$$x_{32} = 56$$
$$x_{33} = 15.707963267949$$
$$x_{34} = 40.8407044966673$$
$$x_{35} = -28.2743338823081$$
$$x_{36} = -62.8318530717959$$
$$x_{37} = -6.28318530717959$$
$$x_{38} = -20$$
$$x_{39} = 3.14159265358979$$
$$x_{40} = -25.1327412287183$$
$$x_{41} = -54$$
$$x_{42} = -84.8230016469244$$
$$x_{43} = -29.75$$
$$x_{44} = 96$$
$$x_{45} = -34.5575191894877$$
$$x_{46} = 0$$
$$x_{47} = 31.4159265358979$$
$$x_{48} = 72$$
$$x_{49} = 47.1238898038469$$
$$x_{50} = 18.8495559215388$$
$$x_{51} = -82$$
$$x_{52} = 84.8230016469244$$
$$x_{53} = 91.106186954104$$
$$x_{54} = 74$$
$$x_{55} = -47.1238898038469$$
$$x_{56} = 65.9734457253857$$
$$x_{57} = -56.5486677646163$$
$$x_{58} = -48$$
$$x_{59} = -64$$
$$x_{60} = 21.9911485751286$$
$$x_{61} = -60$$
$$x_{62} = -32$$
$$x_{63} = 58.25$$
$$x_{64} = 14.25$$
$$x_{65} = 100$$
$$x_{66} = 75.398223686155$$
$$x_{67} = 28$$
$$x_{68} = 87.9645943005142$$
$$x_{69} = -86$$
$$x_{70} = -38$$
$$x_{71} = -3.14159265358979$$
$$x_{72} = 2$$
$$x_{73} = -80$$
$$x_{74} = -12.5663706143592$$
$$x_{75} = -14$$
$$x_{76} = 97.3893722612836$$
$$x_{77} = -100.530964914873$$
$$x_{78} = 59.6902604182061$$
$$x_{79} = -44$$
$$x_{80} = -7.75$$
$$x_{81} = -66$$
$$x_{82} = 43.9822971502571$$
$$x_{83} = -50.2654824574367$$
$$x_{84} = 37.6991118430775$$
$$x_{85} = 18$$
$$x_{86} = -40.8407044966673$$
$$x_{87} = 52$$
$$x_{88} = -4$$
$$x_{89} = -69.1150383789755$$
$$x_{90} = -92$$
$$x_{91} = 9.42477796076938$$
$$x_{92} = 24$$
$$x_{93} = 68$$
$$x_{94} = 62.8318530717959$$
$$x_{95} = -73.75$$
$$x_{96} = 40$$
$$x_{97} = -51.75$$
$$x_{98} = 34$$
$$x_{99} = -42$$
$$x_{100} = 46$$
$$x_{101} = -94.2477796076938$$
so we need to solve the equation:
$$\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -70$$
$$x_{2} = 62$$
$$x_{3} = 90$$
$$x_{4} = 50$$
$$x_{5} = -16$$
$$x_{6} = 94$$
$$x_{7} = -76$$
$$x_{8} = -18.8495559215388$$
$$x_{9} = -36$$
$$x_{10} = 25.1327412287183$$
$$x_{11} = -58$$
$$x_{12} = 8$$
$$x_{13} = 81.6814089933346$$
$$x_{14} = 30$$
$$x_{15} = -22$$
$$x_{16} = 69.1150383789755$$
$$x_{17} = -26$$
$$x_{18} = -88$$
$$x_{19} = -78.5398163397448$$
$$x_{20} = -98$$
$$x_{21} = 36.25$$
$$x_{22} = 6$$
$$x_{23} = -10$$
$$x_{24} = -95.75$$
$$x_{25} = 53.4070751110265$$
$$x_{26} = 84$$
$$x_{27} = 12$$
$$x_{28} = -91.106186954104$$
$$x_{29} = 78$$
$$x_{30} = 80.25$$
$$x_{31} = -72.2566310325652$$
$$x_{32} = 56$$
$$x_{33} = 15.707963267949$$
$$x_{34} = 40.8407044966673$$
$$x_{35} = -28.2743338823081$$
$$x_{36} = -62.8318530717959$$
$$x_{37} = -6.28318530717959$$
$$x_{38} = -20$$
$$x_{39} = 3.14159265358979$$
$$x_{40} = -25.1327412287183$$
$$x_{41} = -54$$
$$x_{42} = -84.8230016469244$$
$$x_{43} = -29.75$$
$$x_{44} = 96$$
$$x_{45} = -34.5575191894877$$
$$x_{46} = 0$$
$$x_{47} = 31.4159265358979$$
$$x_{48} = 72$$
$$x_{49} = 47.1238898038469$$
$$x_{50} = 18.8495559215388$$
$$x_{51} = -82$$
$$x_{52} = 84.8230016469244$$
$$x_{53} = 91.106186954104$$
$$x_{54} = 74$$
$$x_{55} = -47.1238898038469$$
$$x_{56} = 65.9734457253857$$
$$x_{57} = -56.5486677646163$$
$$x_{58} = -48$$
$$x_{59} = -64$$
$$x_{60} = 21.9911485751286$$
$$x_{61} = -60$$
$$x_{62} = -32$$
$$x_{63} = 58.25$$
$$x_{64} = 14.25$$
$$x_{65} = 100$$
$$x_{66} = 75.398223686155$$
$$x_{67} = 28$$
$$x_{68} = 87.9645943005142$$
$$x_{69} = -86$$
$$x_{70} = -38$$
$$x_{71} = -3.14159265358979$$
$$x_{72} = 2$$
$$x_{73} = -80$$
$$x_{74} = -12.5663706143592$$
$$x_{75} = -14$$
$$x_{76} = 97.3893722612836$$
$$x_{77} = -100.530964914873$$
$$x_{78} = 59.6902604182061$$
$$x_{79} = -44$$
$$x_{80} = -7.75$$
$$x_{81} = -66$$
$$x_{82} = 43.9822971502571$$
$$x_{83} = -50.2654824574367$$
$$x_{84} = 37.6991118430775$$
$$x_{85} = 18$$
$$x_{86} = -40.8407044966673$$
$$x_{87} = 52$$
$$x_{88} = -4$$
$$x_{89} = -69.1150383789755$$
$$x_{90} = -92$$
$$x_{91} = 9.42477796076938$$
$$x_{92} = 24$$
$$x_{93} = 68$$
$$x_{94} = 62.8318530717959$$
$$x_{95} = -73.75$$
$$x_{96} = 40$$
$$x_{97} = -51.75$$
$$x_{98} = 34$$
$$x_{99} = -42$$
$$x_{100} = 46$$
$$x_{101} = -94.2477796076938$$
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
$$\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{sign}{\left(\tan{\left(x \right)} \right)} + \tan^{2}{\left(x \right)} + 1 = 0$$
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
$$\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{sign}{\left(\tan{\left(x \right)} \right)} + \tan^{2}{\left(x \right)} + 1 = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \delta\left(\tan{\left(x \right)}\right) + \tan{\left(x \right)} \operatorname{sign}{\left(\tan{\left(x \right)} \right)} + \tan{\left(x \right)}\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right) \delta\left(\tan{\left(x \right)}\right) + \tan{\left(x \right)} \operatorname{sign}{\left(\tan{\left(x \right)} \right)} + \tan{\left(x \right)}\right) = 0$$
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
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x) + Abs(tan(x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|}{x}\right)$$
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:
$$\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right| = - \tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|$$
- No
$$\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right| = \tan{\left(x \right)} - \left|{\tan{\left(x \right)}}\right|$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right| = - \tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right|$$
- No
$$\tan{\left(x \right)} + \left|{\tan{\left(x \right)}}\right| = \tan{\left(x \right)} - \left|{\tan{\left(x \right)}}\right|$$
- No
so, the function
not is
neither even, nor odd