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2 x - 4 f(x) = ------ x

$$f{\left(x \right)} = \frac{x^{2} - 4}{x}$$

f = (x^2 - 4)/x

The graph of the function

The domain of the function

The points at which the function is not precisely defined:

$$x_{1} = 0$$

$$x_{1} = 0$$

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:

$$\frac{x^{2} - 4}{x} = 0$$

Solve this equation

The points of intersection with the axis X:

**Analytical solution**

$$x_{1} = -2$$

$$x_{2} = 2$$

**Numerical solution**

$$x_{1} = -2$$

$$x_{2} = 2$$

so we need to solve the equation:

$$\frac{x^{2} - 4}{x} = 0$$

Solve this equation

The points of intersection with the axis X:

$$x_{1} = -2$$

$$x_{2} = 2$$

$$x_{1} = -2$$

$$x_{2} = 2$$

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:

$$\frac{d}{d x} f{\left(x \right)} = $$

the first derivative

$$2 - \frac{x^{2} - 4}{x^{2}} = 0$$

Solve this equation

Solutions are not found,

function may have no extrema

$$\frac{d}{d x} f{\left(x \right)} = 0$$

(the derivative equals zero),

and the roots of this equation are the extrema of this function:

$$\frac{d}{d x} f{\left(x \right)} = $$

the first derivative

$$2 - \frac{x^{2} - 4}{x^{2}} = 0$$

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:

$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$

the second derivative

$$\frac{2 \left(-1 + \frac{x^{2} - 4}{x^{2}}\right)}{x} = 0$$

Solve this equation

Solutions are not found,

maybe, the function has no inflections

$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$

(the second derivative equals zero),

the roots of this equation will be the inflection points for the specified function graph:

$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$

the second derivative

$$\frac{2 \left(-1 + \frac{x^{2} - 4}{x^{2}}\right)}{x} = 0$$

Solve this equation

Solutions are not found,

maybe, the function has no inflections

Vertical asymptotes

Have:

$$x_{1} = 0$$

$$x_{1} = 0$$

Horizontal asymptotes

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

$$\lim_{x \to -\infty}\left(\frac{x^{2} - 4}{x}\right) = -\infty$$

Let's take the limit

so,

horizontal asymptote on the left doesn’t exist

$$\lim_{x \to \infty}\left(\frac{x^{2} - 4}{x}\right) = \infty$$

Let's take the limit

so,

horizontal asymptote on the right doesn’t exist

$$\lim_{x \to -\infty}\left(\frac{x^{2} - 4}{x}\right) = -\infty$$

Let's take the limit

so,

horizontal asymptote on the left doesn’t exist

$$\lim_{x \to \infty}\left(\frac{x^{2} - 4}{x}\right) = \infty$$

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^2 - 4)/x, divided by x at x->+oo and x ->-oo

$$\lim_{x \to -\infty}\left(\frac{x^{2} - 4}{x^{2}}\right) = 1$$

Let's take the limit

so,

inclined asymptote equation on the left:

$$y = x$$

$$\lim_{x \to \infty}\left(\frac{x^{2} - 4}{x^{2}}\right) = 1$$

Let's take the limit

so,

inclined asymptote equation on the right:

$$y = x$$

$$\lim_{x \to -\infty}\left(\frac{x^{2} - 4}{x^{2}}\right) = 1$$

Let's take the limit

so,

inclined asymptote equation on the left:

$$y = x$$

$$\lim_{x \to \infty}\left(\frac{x^{2} - 4}{x^{2}}\right) = 1$$

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:

$$\frac{x^{2} - 4}{x} = - \frac{x^{2} - 4}{x}$$

- No

$$\frac{x^{2} - 4}{x} = \frac{x^{2} - 4}{x}$$

- No

so, the function

not is

neither even, nor odd

So, check:

$$\frac{x^{2} - 4}{x} = - \frac{x^{2} - 4}{x}$$

- No

$$\frac{x^{2} - 4}{x} = \frac{x^{2} - 4}{x}$$

- No

so, the function

not is

neither even, nor odd