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  • Graphing y =:
  • x^3/2(x+1)^2
  • x^3+6x^2+9x
  • 4x/(x^2+1)^2 4x/(x^2+1)^2
  • -x^4+2x^2+3

  • Identical expressions

  • arcsin(four *x/(four +x^ two))
  • arc sinus of (4 multiply by x divide by (4 plus x squared ))
  • arc sinus of (four multiply by x divide by (four plus x to the power of two))
  • arcsin(4*x/(4+x2))
  • arcsin4*x/4+x2
  • arcsin(4*x/(4+x²))
  • arcsin(4*x/(4+x to the power of 2))
  • arcsin(4x/(4+x^2))
  • arcsin(4x/(4+x2))
  • arcsin4x/4+x2
  • arcsin4x/4+x^2
  • arcsin(4*x divide by (4+x^2))

  • Similar expressions

  • arcsin(4*x/(4-x^2))
Plotting a function graph/ arcsin(4*x/(4+x^2))

Graphing y = arcsin(4*x/(4+x^2))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
           / 4*x  \
f(x) = asin|------|
           |     2|
           \4 + x /
f(x)=asin⁡(4xx2+4)f{\left(x \right)} = \operatorname{asin}{\left(\frac{4 x}{x^{2} + 4} \right)} 
f = asin((4*x)/(x^2 + 4))
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:
asin⁡(4xx2+4)=0\operatorname{asin}{\left(\frac{4 x}{x^{2} + 4} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to asin((4*x)/(4 + x^2)).
asin⁡(0⋅402+4)\operatorname{asin}{\left(\frac{0 \cdot 4}{0^{2} + 4} \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
−8x2(x2+4)2+4x2+4−16x2(x2+4)2+1=0\frac{- \frac{8 x^{2}}{\left(x^{2} + 4\right)^{2}} + \frac{4}{x^{2} + 4}}{\sqrt{- \frac{16 x^{2}}{\left(x^{2} + 4\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
8x(4x2x2+4−3+8(2x2x2+4−1)2(x2+4)(−16x2(x2+4)2+1))(x2+4)2−16x2(x2+4)2+1=0\frac{8 x \left(\frac{4 x^{2}}{x^{2} + 4} - 3 + \frac{8 \left(\frac{2 x^{2}}{x^{2} + 4} - 1\right)^{2}}{\left(x^{2} + 4\right) \left(- \frac{16 x^{2}}{\left(x^{2} + 4\right)^{2}} + 1\right)}\right)}{\left(x^{2} + 4\right)^{2} \sqrt{- \frac{16 x^{2}}{\left(x^{2} + 4\right)^{2}} + 1}} = 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(-\infty, 0\right]
Convex at the intervals
[0,∞)\left[0, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞asin⁡(4xx2+4)=0\lim_{x \to -\infty} \operatorname{asin}{\left(\frac{4 x}{x^{2} + 4} \right)} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞asin⁡(4xx2+4)=0\lim_{x \to \infty} \operatorname{asin}{\left(\frac{4 x}{x^{2} + 4} \right)} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of asin((4*x)/(4 + x^2)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(asin⁡(4xx2+4)x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\frac{4 x}{x^{2} + 4} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(asin⁡(4xx2+4)x)=0\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\frac{4 x}{x^{2} + 4} \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:
asin⁡(4xx2+4)=−asin⁡(4xx2+4)\operatorname{asin}{\left(\frac{4 x}{x^{2} + 4} \right)} = - \operatorname{asin}{\left(\frac{4 x}{x^{2} + 4} \right)}
- No
asin⁡(4xx2+4)=asin⁡(4xx2+4)\operatorname{asin}{\left(\frac{4 x}{x^{2} + 4} \right)} = \operatorname{asin}{\left(\frac{4 x}{x^{2} + 4} \right)}
- No
so, the function
not is
neither even, nor odd
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