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  • Graphing y =:
  • x^4+8x^2+9
  • x^3-12x+1
  • (x^2-x-2)/(x-2)
  • -x/2

  • Identical expressions

  • atan(tan(x/ two)+ one)
  • arc tangent of gent of ( tangent of (x divide by 2) plus 1)
  • arc tangent of gent of ( tangent of (x divide by two) plus one)
  • atantanx/2+1
  • atan(tan(x divide by 2)+1)

  • Similar expressions

  • atan(tan(x/2)-1)
  • arctan(tan(x/2)+1)
Plotting a function graph/ atan(tan(x/2)+1)

Graphing y = atan(tan(x/2)+1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           /   /x\    \
f(x) = atan|tan|-| + 1|
           \   \2/    /
f(x)=atan⁡(tan⁡(x2)+1)f{\left(x \right)} = \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} 
f = atan(tan(x/2) + 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:
atan⁡(tan⁡(x2)+1)=0\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π2x_{1} = - \frac{\pi}{2}
Numerical solution
x1=42.4115008234622x_{1} = 42.4115008234622
x2=54.9778714378214x_{2} = 54.9778714378214
x3=29.845130209103x_{3} = 29.845130209103
x4=−95.8185759344887x_{4} = -95.8185759344887
x5=−64.4026493985908x_{5} = -64.4026493985908
x6=48.6946861306418x_{6} = 48.6946861306418
x7=−32.9867228626928x_{7} = -32.9867228626928
x8=−7.85398163397448x_{8} = -7.85398163397448
x9=105.243353895258x_{9} = 105.243353895258
x10=10.9955742875643x_{10} = 10.9955742875643
x11=−76.9690200129499x_{11} = -76.9690200129499
x12=98.9601685880785x_{12} = 98.9601685880785
x13=306.305283725005x_{13} = 306.305283725005
x14=36.1283155162826x_{14} = 36.1283155162826
x15=23.5619449019235x_{15} = 23.5619449019235
x16=−45.553093477052x_{16} = -45.553093477052
x17=−1.5707963267949x_{17} = -1.5707963267949
x18=−102.101761241668x_{18} = -102.101761241668
x19=67.5442420521806x_{19} = 67.5442420521806
x20=92.6769832808989x_{20} = 92.6769832808989
x21=−58.1194640914112x_{21} = -58.1194640914112
x22=73.8274273593601x_{22} = 73.8274273593601
x23=−39.2699081698724x_{23} = -39.2699081698724
x24=−70.6858347057703x_{24} = -70.6858347057703
x25=80.1106126665397x_{25} = 80.1106126665397
x26=−14.1371669411541x_{26} = -14.1371669411541
x27=−83.2522053201295x_{27} = -83.2522053201295
x28=−89.5353906273091x_{28} = -89.5353906273091
x29=86.3937979737193x_{29} = 86.3937979737193
x30=−26.7035375555132x_{30} = -26.7035375555132
x31=−51.8362787842316x_{31} = -51.8362787842316
x32=17.2787595947439x_{32} = 17.2787595947439
x33=4.71238898038469x_{33} = 4.71238898038469
x34=−20.4203522483337x_{34} = -20.4203522483337
x35=312.588469032184x_{35} = 312.588469032184
x36=61.261056745001x_{36} = 61.261056745001
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to atan(tan(x/2) + 1).
atan⁡(tan⁡(02)+1)\operatorname{atan}{\left(\tan{\left(\frac{0}{2} \right)} + 1 \right)}
The result:
f(0)=π4f{\left(0 \right)} = \frac{\pi}{4}
The point:
(0, pi/4)
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(x2)2+12(tan⁡(x2)+1)2+1=0\frac{\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}}{\left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} + 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
(tan⁡(x2)−(tan⁡(x2)+1)(tan⁡2(x2)+1)(tan⁡(x2)+1)2+1)(tan⁡2(x2)+1)2((tan⁡(x2)+1)2+1)=0\frac{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\left(\tan{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{2 \left(\left(\tan{\left(\frac{x}{2} \right)} + 1\right)^{2} + 1\right)} = 0
Solve this equation
The roots of this equation
x1=−2atan⁡(12−52)x_{1} = - 2 \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}
x2=−2atan⁡(12+52)x_{2} = - 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}

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
(−∞,−2atan⁡(12+52)]∪[−2atan⁡(12−52),∞)\left(-\infty, - 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}\right] \cup \left[- 2 \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}, \infty\right)
Convex at the intervals
[−2atan⁡(12+52),−2atan⁡(12−52)]\left[- 2 \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}, - 2 \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}\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→−∞atan⁡(tan⁡(x2)+1)y = \lim_{x \to -\infty} \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞atan⁡(tan⁡(x2)+1)y = \lim_{x \to \infty} \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(tan(x/2) + 1), 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→−∞(atan⁡(tan⁡(x2)+1)x)y = x \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(atan⁡(tan⁡(x2)+1)x)y = x \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \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:
atan⁡(tan⁡(x2)+1)=−atan⁡(tan⁡(x2)−1)\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} = - \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}
- No
atan⁡(tan⁡(x2)+1)=atan⁡(tan⁡(x2)−1)\operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} = \operatorname{atan}{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}
- No
so, the function
not is
neither even, nor odd
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