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  • Graphing y =:
  • (x^2+5)/(x-3)
  • x^2-|4x+5|
  • x^2+5x+4
  • x^3+6x^2+9x

  • Identical expressions

  • one -atan(x)/ two + three *x/ two +x^ two *atan(x)/ two -x*log(one +x^ two)/ two
  • 1 minus arc tangent of gent of (x) divide by 2 plus 3 multiply by x divide by 2 plus x squared multiply by arc tangent of gent of (x) divide by 2 minus x multiply by logarithm of (1 plus x squared ) divide by 2
  • one minus arc tangent of gent of (x) divide by two plus three multiply by x divide by two plus x to the power of two multiply by arc tangent of gent of (x) divide by two minus x multiply by logarithm of (one plus x to the power of two) divide by two
  • 1-atan(x)/2+3*x/2+x2*atan(x)/2-x*log(1+x2)/2
  • 1-atanx/2+3*x/2+x2*atanx/2-x*log1+x2/2
  • 1-atan(x)/2+3*x/2+x²*atan(x)/2-x*log(1+x²)/2
  • 1-atan(x)/2+3*x/2+x to the power of 2*atan(x)/2-x*log(1+x to the power of 2)/2
  • 1-atan(x)/2+3x/2+x^2atan(x)/2-xlog(1+x^2)/2
  • 1-atan(x)/2+3x/2+x2atan(x)/2-xlog(1+x2)/2
  • 1-atanx/2+3x/2+x2atanx/2-xlog1+x2/2
  • 1-atanx/2+3x/2+x^2atanx/2-xlog1+x^2/2
  • 1-atan(x) divide by 2+3*x divide by 2+x^2*atan(x) divide by 2-x*log(1+x^2) divide by 2

  • Similar expressions

  • 1-atan(x)/2+3*x/2+x^2*atan(x)/2-x*log(1-x^2)/2
  • 1+atan(x)/2+3*x/2+x^2*atan(x)/2-x*log(1+x^2)/2
  • 1-atan(x)/2+3*x/2-x^2*atan(x)/2-x*log(1+x^2)/2
  • 1-atan(x)/2+3*x/2+x^2*atan(x)/2+x*log(1+x^2)/2
  • 1-atan(x)/2-3*x/2+x^2*atan(x)/2-x*log(1+x^2)/2
  • 1-arctan(x)/2+3*x/2+x^2*arctan(x)/2-x*log(1+x^2)/2
  • 1-arctanx/2+3*x/2+x^2*arctanx/2-x*log(1+x^2)/2
Plotting a function graph/ 1-atan(x)/2+3*x/2+x^2*atan(x)/2-x*log(1+x^2)/2

Graphing y = 1-atan(x)/2+3*x/2+x^2*atan(x)/2-x*log(1+x^2)/2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                            2                /     2\
           atan(x)   3*x   x *atan(x)   x*log\1 + x /
f(x) = 1 - ------- + --- + ---------- - -------------
              2       2        2              2
$$f{\left(x \right)} = - \frac{x \log{\left(x^{2} + 1 \right)}}{2} + \left(\frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} + \left(\frac{3 x}{2} + \left(- \frac{\operatorname{atan}{\left(x \right)}}{2} + 1\right)\right)\right)$$ 
f = -x*log(x^2 + 1)/2 + (x^2*atan(x))/2 + (3*x)/2 - atan(x)/2 + 1
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:
$$- \frac{x \log{\left(x^{2} + 1 \right)}}{2} + \left(\frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} + \left(\frac{3 x}{2} + \left(- \frac{\operatorname{atan}{\left(x \right)}}{2} + 1\right)\right)\right) = 0$$
Solve this equation
The points of intersection with the axis X:

Numerical solution
$$x_{1} = -0.890152452384207$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 1 - atan(x)/2 + (3*x)/2 + (x^2*atan(x))/2 - x*log(1 + x^2)/2.
$$- \frac{0 \log{\left(0^{2} + 1 \right)}}{2} + \left(\frac{0^{2} \operatorname{atan}{\left(0 \right)}}{2} + \left(\frac{0 \cdot 3}{2} + \left(- \frac{\operatorname{atan}{\left(0 \right)}}{2} + 1\right)\right)\right)$$
The result:
$$f{\left(0 \right)} = 1$$
The point:
(0, 1)
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
$$- \frac{x^{2}}{2 \left(x^{2} + 1\right)} + x \operatorname{atan}{\left(x \right)} - \frac{\log{\left(x^{2} + 1 \right)}}{2} + \frac{3}{2} - \frac{1}{2 \left(x^{2} + 1\right)} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{x^{3}}{\left(x^{2} + 1\right)^{2}} - \frac{x}{x^{2} + 1} + \frac{x}{\left(x^{2} + 1\right)^{2}} + \operatorname{atan}{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$

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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(- \frac{x \log{\left(x^{2} + 1 \right)}}{2} + \left(\frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} + \left(\frac{3 x}{2} + \left(- \frac{\operatorname{atan}{\left(x \right)}}{2} + 1\right)\right)\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(- \frac{x \log{\left(x^{2} + 1 \right)}}{2} + \left(\frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} + \left(\frac{3 x}{2} + \left(- \frac{\operatorname{atan}{\left(x \right)}}{2} + 1\right)\right)\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 1 - atan(x)/2 + (3*x)/2 + (x^2*atan(x))/2 - x*log(1 + x^2)/2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{- \frac{x \log{\left(x^{2} + 1 \right)}}{2} + \left(\frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} + \left(\frac{3 x}{2} + \left(- \frac{\operatorname{atan}{\left(x \right)}}{2} + 1\right)\right)\right)}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{- \frac{x \log{\left(x^{2} + 1 \right)}}{2} + \left(\frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} + \left(\frac{3 x}{2} + \left(- \frac{\operatorname{atan}{\left(x \right)}}{2} + 1\right)\right)\right)}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$- \frac{x \log{\left(x^{2} + 1 \right)}}{2} + \left(\frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} + \left(\frac{3 x}{2} + \left(- \frac{\operatorname{atan}{\left(x \right)}}{2} + 1\right)\right)\right) = - \frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} + \frac{x \log{\left(x^{2} + 1 \right)}}{2} - \frac{3 x}{2} + \frac{\operatorname{atan}{\left(x \right)}}{2} + 1$$
- No
$$- \frac{x \log{\left(x^{2} + 1 \right)}}{2} + \left(\frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} + \left(\frac{3 x}{2} + \left(- \frac{\operatorname{atan}{\left(x \right)}}{2} + 1\right)\right)\right) = \frac{x^{2} \operatorname{atan}{\left(x \right)}}{2} - \frac{x \log{\left(x^{2} + 1 \right)}}{2} + \frac{3 x}{2} - \frac{\operatorname{atan}{\left(x \right)}}{2} - 1$$
- No
so, the function
not is
neither even, nor odd
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