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Plotting a function graph/ y=tgx-2

Graphing y = y=tgx-2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = tan(x) - 2
f(x)=tan⁡(x)−2f{\left(x \right)} = \tan{\left(x \right)} - 2 
f = tan(x) - 2
The graph of the function
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.0-2000020000
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⁡(x)−2=0\tan{\left(x \right)} - 2 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=atan⁡(2)x_{1} = \operatorname{atan}{\left(2 \right)}
Numerical solution
x1=−61.7247043540018x_{1} = -61.7247043540018
x2=−17.7424072037447x_{2} = -17.7424072037447
x3=67.0805944431797x_{3} = 67.0805944431797
x4=−46.0167410860528x_{4} = -46.0167410860528
x5=−52.2999263932324x_{5} = -52.2999263932324
x6=−55.4415190468222x_{6} = -55.4415190468222
x7=−1258.67150537171x_{7} = -1258.67150537171
x8=−74.2910749683609x_{8} = -74.2910749683609
x9=−30.3087778181038x_{9} = -30.3087778181038
x10=−24.0255925109243x_{10} = -24.0255925109243
x11=101.638113632667x_{11} = 101.638113632667
x12=−96.2822235434895x_{12} = -96.2822235434895
x13=76.5053724039491x_{13} = 76.5053724039491
x14=−39.7335557788732x_{14} = -39.7335557788732
x15=10.5319266785635x_{15} = 10.5319266785635
x16=63.93900178959x_{16} = 63.93900178959
x17=−8.31762924297529x_{17} = -8.31762924297529
x18=51.3726311752308x_{18} = 51.3726311752308
x19=−33.4503704716936x_{19} = -33.4503704716936
x20=54.5142238288206x_{20} = 54.5142238288206
x21=92.2133356718981x_{21} = 92.2133356718981
x22=41.9478532144614x_{22} = 41.9478532144614
x23=−49.1583337396426x_{23} = -49.1583337396426
x24=−36.5919631252834x_{24} = -36.5919631252834
x25=−102.565408850669x_{25} = -102.565408850669
x26=23.0982972929226x_{26} = 23.0982972929226
x27=−99.4238161970793x_{27} = -99.4238161970793
x28=35.6646679072818x_{28} = 35.6646679072818
x29=19.9567046393328x_{29} = 19.9567046393328
x30=4.24874137138388x_{30} = 4.24874137138388
x31=98.4965209790777x_{31} = 98.4965209790777
x32=85.9301503647185x_{32} = 85.9301503647185
x33=−89.9990382363099x_{33} = -89.9990382363099
x34=48.231038521641x_{34} = 48.231038521641
x35=73.3637797503593x_{35} = 73.3637797503593
x36=70.2221870967695x_{36} = 70.2221870967695
x37=1.10714871779409x_{37} = 1.10714871779409
x38=−27.167185164514x_{38} = -27.167185164514
x39=16.8151119857431x_{39} = 16.8151119857431
x40=32.523075253692x_{40} = 32.523075253692
x41=13.6735193321533x_{41} = 13.6735193321533
x42=79.6469650575389x_{42} = 79.6469650575389
x43=−11.4592218965651x_{43} = -11.4592218965651
x44=−77.4326676219507x_{44} = -77.4326676219507
x45=26.2398899465124x_{45} = 26.2398899465124
x46=60.7974091360002x_{46} = 60.7974091360002
x47=−68.0078896611814x_{47} = -68.0078896611814
x48=95.3549283254879x_{48} = 95.3549283254879
x49=−2.0344439357957x_{49} = -2.0344439357957
x50=−83.7158529291303x_{50} = -83.7158529291303
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x) - 2.
−2+tan⁡(0)-2 + \tan{\left(0 \right)}
The result:
f(0)=−2f{\left(0 \right)} = -2
The point:
(0, -2)
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
tan⁡2(x)+1=0\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
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
2(tan⁡2(x)+1)tan⁡(x)=02 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0

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[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(tan⁡(x)−2)y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} - 2\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(tan⁡(x)−2)y = \lim_{x \to \infty}\left(\tan{\left(x \right)} - 2\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x) - 2, divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(tan⁡(x)−2x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} - 2}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(x)−2x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} - 2}{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⁡(x)−2=−tan⁡(x)−2\tan{\left(x \right)} - 2 = - \tan{\left(x \right)} - 2
- No
tan⁡(x)−2=tan⁡(x)+2\tan{\left(x \right)} - 2 = \tan{\left(x \right)} + 2
- No
so, the function
not is
neither even, nor odd
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