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

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       x - 3
f(x) = -----
       x - 2

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

f = (x - 3)/(x - 2)

The graph of the function

The domain of the function

The points at which the function is not precisely defined:

x_{1} = 2

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 - 3}{x - 2} = 0

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

Analytical solution

x_{1} = 3

Numerical solution

x_{1} = 3

The points of intersection with the Y axis coordinate

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

- \frac{3}{-2}

The result:

f{\left(0 \right)} = \frac{3}{2}

The point:

(0, 3/2)

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)} =

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

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

Vertical asymptotes

Have:

x_{1} = 2

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

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

y = 1

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

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

y = 1

Inclined asymptotes

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

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

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

\lim_{x \to \infty}\left(\frac{x - 3}{x \left(x - 2\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{x - 3}{x - 2} = \frac{- x - 3}{- x - 2}

- No

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

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