Graphing y = (-2x+1)/(x+3)

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       -2*x + 1
f(x) = --------
        x + 3

f{\left(x \right)} = \frac{1 - 2 x}{x + 3}

f = (1 - 2*x)/(x + 3)

The graph of the function

The domain of the function

The points at which the function is not precisely defined:

x_{1} = -3

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{1 - 2 x}{x + 3} = 0

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

Analytical solution

x_{1} = \frac{1}{2}

Numerical solution

x_{1} = 0.5

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to (-2*x + 1)/(x + 3).

\frac{1 - 0}{3}

The result:

f{\left(0 \right)} = \frac{1}{3}

The point:

(0, 1/3)

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{1 - 2 x}{\left(x + 3\right)^{2}} - \frac{2}{x + 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 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{2 \left(2 - \frac{2 x - 1}{x + 3}\right)}{\left(x + 3\right)^{2}} = 0

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

Vertical asymptotes

Have:

x_{1} = -3

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{1 - 2 x}{x + 3}\right) = -2

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

y = -2

\lim_{x \to \infty}\left(\frac{1 - 2 x}{x + 3}\right) = -2

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

y = -2

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of (-2*x + 1)/(x + 3), divided by x at x->+oo and x ->-oo

\lim_{x \to -\infty}\left(\frac{1 - 2 x}{x \left(x + 3\right)}\right) = 0

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

\lim_{x \to \infty}\left(\frac{1 - 2 x}{x \left(x + 3\right)}\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:

\frac{1 - 2 x}{x + 3} = \frac{2 x + 1}{3 - x}

- No

\frac{1 - 2 x}{x + 3} = - \frac{2 x + 1}{3 - x}

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