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  • How to use it?

  • Graphing y =:
  • x^2+4x+5
  • x^2+2x+4
  • -x^2-2x+3
  • -x^2+2x-3
  • Derivative of:
  • x/sqrt(x^2+1) x/sqrt(x^2+1)
  • Integral of d{x}:
  • x/sqrt(x^2+1) x/sqrt(x^2+1)

  • Identical expressions

  • x/sqrt(x^ two + one)
  • x divide by square root of (x squared plus 1)
  • x divide by square root of (x to the power of two plus one)
  • x/√(x^2+1)
  • x/sqrt(x2+1)
  • x/sqrtx2+1
  • x/sqrt(x²+1)
  • x/sqrt(x to the power of 2+1)
  • x/sqrtx^2+1
  • x divide by sqrt(x^2+1)

  • Similar expressions

  • x/sqrt(x^2-1)
Plotting a function graph/ x/sqrt(x^2+1)

Graphing y = x/sqrt(x^2+1)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            x     
f(x) = -----------
          ________
         /  2     
       \/  x  + 1
f(x)=xx2+1f{\left(x \right)} = \frac{x}{\sqrt{x^{2} + 1}} 
f = x/sqrt(x^2 + 1)
The graph of the function
02468-8-6-4-2-10102-2
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:
xx2+1=0\frac{x}{\sqrt{x^{2} + 1}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
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 x/sqrt(x^2 + 1).
002+1\frac{0}{\sqrt{0^{2} + 1}}
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
−x2(x2+1)32+1x2+1=0- \frac{x^{2}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + \frac{1}{\sqrt{x^{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
x(3x2x2+1−3)(x2+1)32=0\frac{x \left(\frac{3 x^{2}}{x^{2} + 1} - 3\right)}{\left(x^{2} + 1\right)^{\frac{3}{2}}} = 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→−∞(xx2+1)=−1\lim_{x \to -\infty}\left(\frac{x}{\sqrt{x^{2} + 1}}\right) = -1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−1y = -1
lim⁡x→∞(xx2+1)=1\lim_{x \to \infty}\left(\frac{x}{\sqrt{x^{2} + 1}}\right) = 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=1y = 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/sqrt(x^2 + 1), divided by x at x->+oo and x ->-oo
lim⁡x→−∞1x2+1=0\lim_{x \to -\infty} \frac{1}{\sqrt{x^{2} + 1}} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞1x2+1=0\lim_{x \to \infty} \frac{1}{\sqrt{x^{2} + 1}} = 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:
xx2+1=−xx2+1\frac{x}{\sqrt{x^{2} + 1}} = - \frac{x}{\sqrt{x^{2} + 1}}
- No
xx2+1=xx2+1\frac{x}{\sqrt{x^{2} + 1}} = \frac{x}{\sqrt{x^{2} + 1}}
- No
so, the function
not is
neither even, nor odd
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