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Plotting a function graph/ x/5-5/x

Graphing y = x/5-5/x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       x   5
f(x) = - - -
       5   x
f(x)=x5−5xf{\left(x \right)} = \frac{x}{5} - \frac{5}{x} 
f = x/5 - 5/x
The graph of the function
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.0-2000020000
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
x5−5x=0\frac{x}{5} - \frac{5}{x} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−5x_{1} = -5
x2=5x_{2} = 5
Numerical solution
x1=−5x_{1} = -5
x2=5x_{2} = 5
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x/5 - 5/x.
05−50\frac{0}{5} - \frac{5}{0}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
15+5x2=0\frac{1}{5} + \frac{5}{x^{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
−10x3=0- \frac{10}{x^{3}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(x5−5x)=−∞\lim_{x \to -\infty}\left(\frac{x}{5} - \frac{5}{x}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(x5−5x)=∞\lim_{x \to \infty}\left(\frac{x}{5} - \frac{5}{x}\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 x/5 - 5/x, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(x5−5xx)=15\lim_{x \to -\infty}\left(\frac{\frac{x}{5} - \frac{5}{x}}{x}\right) = \frac{1}{5}
Let's take the limit
so,
inclined asymptote equation on the left:
y=x5y = \frac{x}{5}
lim⁡x→∞(x5−5xx)=15\lim_{x \to \infty}\left(\frac{\frac{x}{5} - \frac{5}{x}}{x}\right) = \frac{1}{5}
Let's take the limit
so,
inclined asymptote equation on the right:
y=x5y = \frac{x}{5}
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:
x5−5x=−x5+5x\frac{x}{5} - \frac{5}{x} = - \frac{x}{5} + \frac{5}{x}
- No
x5−5x=x5−5x\frac{x}{5} - \frac{5}{x} = \frac{x}{5} - \frac{5}{x}
- No
so, the function
not is
neither even, nor odd
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