Graphing y = 1/(y^3)

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         1 
f(y) = 1*--
          3
         y

f{\left(y \right)} = 1 \cdot \frac{1}{y^{3}}

f = 1/y^3

The graph of the function

The domain of the function

The points at which the function is not precisely defined:

y_{1} = 0

The points of intersection with the X-axis coordinate

Graph of the function intersects the axis Y at f = 0
so we need to solve the equation:

1 \cdot \frac{1}{y^{3}} = 0

Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis Y

The points of intersection with the Y axis coordinate

The graph crosses Y axis when y equals 0:
substitute y = 0 to 1/y^3.

1 \cdot \frac{1}{0^{3}}

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 y} f{\left(y \right)} = 0

(the derivative equals zero),
and the roots of this equation are the extrema of this function:

\frac{d}{d y} f{\left(y \right)} =

- \frac{3}{y y^{3}} = 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 y^{2}} f{\left(y \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 y^{2}} f{\left(y \right)} =

\frac{12}{y^{5}} = 0

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

Vertical asymptotes

Have:

y_{1} = 0

Horizontal asymptotes

Let’s find horizontal asymptotes with help of the limits of this function at y->+oo and y->-oo

\lim_{y \to -\infty}\left(1 \cdot \frac{1}{y^{3}}\right) = 0

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

y = 0

\lim_{y \to \infty}\left(1 \cdot \frac{1}{y^{3}}\right) = 0

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

y = 0

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of 1/y^3, divided by y at y->+oo and y ->-oo

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

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

\lim_{y \to \infty}\left(\frac{1}{y y^{3}}\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(-y) и f = -f(-y).
So, check:

1 \cdot \frac{1}{y^{3}} = - \frac{1}{y^{3}}

- No

1 \cdot \frac{1}{y^{3}} = \frac{1}{y^{3}}

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

The graph