Graphing y = x-(9/x)

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           9
f(x) = x - -
           x

f{\left(x \right)} = x - \frac{9}{x}

f = x - 9/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:

x - \frac{9}{x} = 0

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

Analytical solution

x_{1} = -3

x_{2} = 3

Numerical solution

x_{1} = 3

x_{2} = -3

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to x - 9/x.

0 - \frac{9}{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)} =

1 + \frac{9}{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

\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{18}{x^{3}} = 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(x - \frac{9}{x}\right) = -\infty

Let's take the limit
so,
horizontal asymptote on the left doesn’t exist

\lim_{x \to \infty}\left(x - \frac{9}{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 - 9/x, divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{x - \frac{9}{x}}{x}\right) = 1

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

y = x

\lim_{x \to \infty}\left(\frac{x - \frac{9}{x}}{x}\right) = 1

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

y = x

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:

x - \frac{9}{x} = - x + \frac{9}{x}

- No

x - \frac{9}{x} = x - \frac{9}{x}

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

The graph