Mister Exam
Graphing y = |x+2|/(x-2)
The solution
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|x + 2|
f(x) = -------
x - 2
$$f{\left(x \right)} = \frac{\left|{x + 2}\right|}{x - 2}$$
f = |x + 2|/(x - 2)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 2$$
$$x_{1} = 2$$
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:
$$\frac{\left|{x + 2}\right|}{x - 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
Numerical solution
$$x_{1} = -2$$
so we need to solve the equation:
$$\frac{\left|{x + 2}\right|}{x - 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
Numerical solution
$$x_{1} = -2$$
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
$$\frac{\operatorname{sign}{\left(x + 2 \right)}}{x - 2} - \frac{\left|{x + 2}\right|}{\left(x - 2\right)^{2}} = 0$$
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
$$\frac{\operatorname{sign}{\left(x + 2 \right)}}{x - 2} - \frac{\left|{x + 2}\right|}{\left(x - 2\right)^{2}} = 0$$
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
$$\frac{2 \left(\delta\left(x + 2\right) - \frac{\operatorname{sign}{\left(x + 2 \right)}}{x - 2} + \frac{\left|{x + 2}\right|}{\left(x - 2\right)^{2}}\right)}{x - 2} = 0$$
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
$$\frac{2 \left(\delta\left(x + 2\right) - \frac{\operatorname{sign}{\left(x + 2 \right)}}{x - 2} + \frac{\left|{x + 2}\right|}{\left(x - 2\right)^{2}}\right)}{x - 2} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 2$$
$$x_{1} = 2$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{x + 2}\right|}{x - 2}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(\frac{\left|{x + 2}\right|}{x - 2}\right) = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty}\left(\frac{\left|{x + 2}\right|}{x - 2}\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(\frac{\left|{x + 2}\right|}{x - 2}\right) = 1$$
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 + 2|/(x - 2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{x + 2}\right|}{x \left(x - 2\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left|{x + 2}\right|}{x \left(x - 2\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left|{x + 2}\right|}{x \left(x - 2\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left|{x + 2}\right|}{x \left(x - 2\right)}\right) = 0$$
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:
$$\frac{\left|{x + 2}\right|}{x - 2} = \frac{\left|{x - 2}\right|}{- x - 2}$$
- No
$$\frac{\left|{x + 2}\right|}{x - 2} = - \frac{\left|{x - 2}\right|}{- x - 2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\left|{x + 2}\right|}{x - 2} = \frac{\left|{x - 2}\right|}{- x - 2}$$
- No
$$\frac{\left|{x + 2}\right|}{x - 2} = - \frac{\left|{x - 2}\right|}{- x - 2}$$
- No
so, the function
not is
neither even, nor odd