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sqrt(arctan(sqrt(2x+3)))
Plotting a function graph/ sqrt(arctan(sqrt(2x+3)))

Graphing y = sqrt(arctan(sqrt(2x+3)))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = \/  atan\\/ 2*x + 3 /
f(x)=atan⁡(2x+3)f{\left(x \right)} = \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} 
f = sqrt(atan(sqrt(2*x + 3)))
The graph of the function
02468-8-6-4-2-101002
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⁡(2x+3)=0\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−32x_{1} = - \frac{3}{2}
Numerical solution
x1=−1.5x_{1} = -1.5
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(atan(sqrt(2*x + 3))).
atan⁡(2⋅0+3)\sqrt{\operatorname{atan}{\left(\sqrt{2 \cdot 0 + 3} \right)}}
The result:
f(0)=3π3f{\left(0 \right)} = \frac{\sqrt{3} \sqrt{\pi}}{3}
The point:
(0, sqrt(3)*sqrt(pi)/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
122x+3⋅(2x+4)atan⁡(2x+3)=0\frac{1}{2 \sqrt{2 x + 3} \cdot \left(2 x + 4\right) \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}} = 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
−4(x+2)2x+3+4(2x+3)32+1(x+2)(2x+3)atan⁡(2x+3)16(x+2)atan⁡(2x+3)=0- \frac{\frac{4}{\left(x + 2\right) \sqrt{2 x + 3}} + \frac{4}{\left(2 x + 3\right)^{\frac{3}{2}}} + \frac{1}{\left(x + 2\right) \left(2 x + 3\right) \operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}}{16 \left(x + 2\right) \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}} = 0
Solve this equation
The roots of this equation
x1=107577.56403671x_{1} = 107577.56403671
x2=62903.8628749537x_{2} = 62903.8628749537
x3=60555.049520562x_{3} = 60555.049520562
x4=79353.594352919x_{4} = 79353.594352919
x5=67602.4245400132x_{5} = 67602.4245400132
x6=116990.63189526x_{6} = 116990.63189526
x7=86407.0907612323x_{7} = 86407.0907612323
x8=39433.7739186636x_{8} = 39433.7739186636
x9=105224.644640647x_{9} = 105224.644640647
x10=109930.626877423x_{10} = 109930.626877423
x11=74652.3743804977x_{11} = 74652.3743804977
x12=77002.8666831961x_{12} = 77002.8666831961
x13=69952.1408924897x_{13} = 69952.1408924897
x14=32403.2572766674x_{14} = 32403.2572766674
x15=51163.3629419719x_{15} = 51163.3629419719
x16=37089.5579402752x_{16} = 37089.5579402752
x17=95814.5013977822x_{17} = 95814.5013977822
x18=58206.5716651308x_{18} = 58206.5716651308
x19=102871.873473385x_{19} = 102871.873473385
x20=55858.4494176989x_{20} = 55858.4494176989
x21=46469.998895868x_{21} = 46469.998895868
x22=48816.4507038095x_{22} = 48816.4507038095
x23=98166.7963429332x_{23} = 98166.7963429332
x24=88758.6644284932x_{24} = 88758.6644284932
x25=72302.1284767751x_{25} = 72302.1284767751
x26=114637.164996904x_{26} = 114637.164996904
x27=44124.041963804x_{27} = 44124.041963804
x28=84055.7155798312x_{28} = 84055.7155798312
x29=41778.6188611969x_{29} = 41778.6188611969
x30=93462.376769134x_{30} = 93462.376769134
x31=112283.828630407x_{31} = 112283.828630407
x32=91110.4288464257x_{32} = 91110.4288464257
x33=34746.0296041462x_{33} = 34746.0296041462
x34=81704.5471562711x_{34} = 81704.5471562711
x35=100519.255590868x_{35} = 100519.255590868
x36=65252.9934430143x_{36} = 65252.9934430143
x37=53510.7049758468x_{37} = 53510.7049758468
x38=119344.225445893x_{38} = 119344.225445893

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:
Have no bends at the whole real axis
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞atan⁡(2x+3)=2π2\lim_{x \to -\infty} \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = \frac{\sqrt{2} \sqrt{\pi}}{2}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=2π2y = \frac{\sqrt{2} \sqrt{\pi}}{2}
lim⁡x→∞atan⁡(2x+3)=2π2\lim_{x \to \infty} \sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = \frac{\sqrt{2} \sqrt{\pi}}{2}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=2π2y = \frac{\sqrt{2} \sqrt{\pi}}{2}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(atan(sqrt(2*x + 3))), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(atan⁡(2x+3)x)=0\lim_{x \to -\infty}\left(\frac{\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(atan⁡(2x+3)x)=0\lim_{x \to \infty}\left(\frac{\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \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:
atan⁡(2x+3)=atan⁡(−2x+3)\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = \sqrt{\operatorname{atan}{\left(\sqrt{- 2 x + 3} \right)}}
- No
atan⁡(2x+3)=−atan⁡(−2x+3)\sqrt{\operatorname{atan}{\left(\sqrt{2 x + 3} \right)}} = - \sqrt{\operatorname{atan}{\left(\sqrt{- 2 x + 3} \right)}}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sqrt(arctan(sqrt(2x+3)))
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