Graphing y = x^-5/x-3

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        1      
f(x) = ---- - 3
        5      
       x *x

f{\left(x \right)} = -3 + \frac{1}{x x^{5}}

f = -3 + 1/(x^5*x)

The graph of the function

The domain of the function

The points at which the function is not precisely defined:

x_{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:

-3 + \frac{1}{x x^{5}} = 0

Solve this equation
The points of intersection with the axis X:

Analytical solution

x_{1} = - \frac{3^{\frac{5}{6}}}{3}

x_{2} = \frac{3^{\frac{5}{6}}}{3}

Numerical solution

x_{1} = -0.832683177655604

x_{2} = 0.832683177655604

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to 1/(x^5*x) - 3.

-3 + \frac{1}{0 \cdot 0}

The result:

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

\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)} =

- \frac{6}{x^{7}} = 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)} =

\frac{42}{x^{8}} = 0

Solve this equation
Solutions are not found,
maybe, the function has no inflections

Vertical asymptotes

Have:

x_{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 \to -\infty}\left(-3 + \frac{1}{x x^{5}}\right) = -3

Let's take the limit
so,
equation of the horizontal asymptote on the left:

y = -3

\lim_{x \to \infty}\left(-3 + \frac{1}{x x^{5}}\right) = -3

Let's take the limit
so,
equation of the horizontal asymptote on the right:

y = -3

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of 1/(x^5*x) - 3, divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{-3 + \frac{1}{x x^{5}}}{x}\right) = 0

Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right

\lim_{x \to \infty}\left(\frac{-3 + \frac{1}{x x^{5}}}{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:

-3 + \frac{1}{x x^{5}} = -3 + \frac{1}{x^{6}}

- No

-3 + \frac{1}{x x^{5}} = 3 - \frac{1}{x^{6}}

- No
so, the function
not is
neither even, nor odd