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Plotting a function graph/ x/(x^2-2)

Graphing y = x/(x^2-2)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
         x   
f(x) = ------
        2    
       x  - 2
f(x)=xx2−2f{\left(x \right)} = \frac{x}{x^{2} - 2} 
f = x/(x^2 - 2)
The graph of the function
0.003.000.250.500.751.001.251.501.752.002.252.502.75-50005000
The domain of the function
The points at which the function is not precisely defined:
x1=−1.4142135623731x_{1} = -1.4142135623731
x2=1.4142135623731x_{2} = 1.4142135623731
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:
xx2−2=0\frac{x}{x^{2} - 2} = 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/(x^2 - 2).
0−2+02\frac{0}{-2 + 0^{2}}
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
−2x2(x2−2)2+1x2−2=0- \frac{2 x^{2}}{\left(x^{2} - 2\right)^{2}} + \frac{1}{x^{2} - 2} = 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
2x(4x2x2−2−3)(x2−2)2=0\frac{2 x \left(\frac{4 x^{2}}{x^{2} - 2} - 3\right)}{\left(x^{2} - 2\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=−1.4142135623731x_{1} = -1.4142135623731
x2=1.4142135623731x_{2} = 1.4142135623731

lim⁡x→−1.4142135623731−(2x(4x2x2−2−3)(x2−2)2)=−2.58359429617981⋅1047\lim_{x \to -1.4142135623731^-}\left(\frac{2 x \left(\frac{4 x^{2}}{x^{2} - 2} - 3\right)}{\left(x^{2} - 2\right)^{2}}\right) = -2.58359429617981 \cdot 10^{47}
lim⁡x→−1.4142135623731+(2x(4x2x2−2−3)(x2−2)2)=−2.58359429617981⋅1047\lim_{x \to -1.4142135623731^+}\left(\frac{2 x \left(\frac{4 x^{2}}{x^{2} - 2} - 3\right)}{\left(x^{2} - 2\right)^{2}}\right) = -2.58359429617981 \cdot 10^{47}
- limits are equal, then skip the corresponding point
lim⁡x→1.4142135623731−(2x(4x2x2−2−3)(x2−2)2)=2.58359429617981⋅1047\lim_{x \to 1.4142135623731^-}\left(\frac{2 x \left(\frac{4 x^{2}}{x^{2} - 2} - 3\right)}{\left(x^{2} - 2\right)^{2}}\right) = 2.58359429617981 \cdot 10^{47}
lim⁡x→1.4142135623731+(2x(4x2x2−2−3)(x2−2)2)=2.58359429617981⋅1047\lim_{x \to 1.4142135623731^+}\left(\frac{2 x \left(\frac{4 x^{2}}{x^{2} - 2} - 3\right)}{\left(x^{2} - 2\right)^{2}}\right) = 2.58359429617981 \cdot 10^{47}
- limits are equal, then skip the corresponding point

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)
Vertical asymptotes
Have:
x1=−1.4142135623731x_{1} = -1.4142135623731
x2=1.4142135623731x_{2} = 1.4142135623731
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(xx2−2)=0\lim_{x \to -\infty}\left(\frac{x}{x^{2} - 2}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞(xx2−2)=0\lim_{x \to \infty}\left(\frac{x}{x^{2} - 2}\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 x/(x^2 - 2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞1x2−2=0\lim_{x \to -\infty} \frac{1}{x^{2} - 2} = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞1x2−2=0\lim_{x \to \infty} \frac{1}{x^{2} - 2} = 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:
xx2−2=−xx2−2\frac{x}{x^{2} - 2} = - \frac{x}{x^{2} - 2}
- No
xx2−2=xx2−2\frac{x}{x^{2} - 2} = \frac{x}{x^{2} - 2}
- Yes
so, the function
is
odd
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