Mister Exam
Graphing y = f(x)=|x+2|/arctg(x+2)
The solution
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|x + 2|
f(x) = -----------
atan(x + 2)
$$f{\left(x \right)} = \frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}}$$
f = |x + 2|/atan(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:
$$\frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\frac{\operatorname{sign}{\left(x + 2 \right)}}{\operatorname{atan}{\left(x + 2 \right)}} - \frac{\left|{x + 2}\right|}{\left(\left(x + 2\right)^{2} + 1\right) \operatorname{atan}^{2}{\left(x + 2 \right)}} = 0$$
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
$$\frac{\operatorname{sign}{\left(x + 2 \right)}}{\operatorname{atan}{\left(x + 2 \right)}} - \frac{\left|{x + 2}\right|}{\left(\left(x + 2\right)^{2} + 1\right) \operatorname{atan}^{2}{\left(x + 2 \right)}} = 0$$
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
$$\frac{2 \left(\delta\left(x + 2\right) - \frac{\operatorname{sign}{\left(x + 2 \right)}}{\left(\left(x + 2\right)^{2} + 1\right) \operatorname{atan}{\left(x + 2 \right)}} + \frac{\left(x + 2 + \frac{1}{\operatorname{atan}{\left(x + 2 \right)}}\right) \left|{x + 2}\right|}{\left(\left(x + 2\right)^{2} + 1\right)^{2} \operatorname{atan}{\left(x + 2 \right)}}\right)}{\operatorname{atan}{\left(x + 2 \right)}} = 0$$
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
$$\frac{2 \left(\delta\left(x + 2\right) - \frac{\operatorname{sign}{\left(x + 2 \right)}}{\left(\left(x + 2\right)^{2} + 1\right) \operatorname{atan}{\left(x + 2 \right)}} + \frac{\left(x + 2 + \frac{1}{\operatorname{atan}{\left(x + 2 \right)}}\right) \left|{x + 2}\right|}{\left(\left(x + 2\right)^{2} + 1\right)^{2} \operatorname{atan}{\left(x + 2 \right)}}\right)}{\operatorname{atan}{\left(x + 2 \right)}} = 0$$
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
$$\lim_{x \to -\infty}\left(\frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}}\right) = \infty$$
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|/atan(x + 2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{x + 2}\right|}{x \operatorname{atan}{\left(x + 2 \right)}}\right) = \frac{2}{\pi}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{2 x}{\pi}$$
$$\lim_{x \to \infty}\left(\frac{\left|{x + 2}\right|}{x \operatorname{atan}{\left(x + 2 \right)}}\right) = \frac{2}{\pi}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = \frac{2 x}{\pi}$$
$$\lim_{x \to -\infty}\left(\frac{\left|{x + 2}\right|}{x \operatorname{atan}{\left(x + 2 \right)}}\right) = \frac{2}{\pi}$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = \frac{2 x}{\pi}$$
$$\lim_{x \to \infty}\left(\frac{\left|{x + 2}\right|}{x \operatorname{atan}{\left(x + 2 \right)}}\right) = \frac{2}{\pi}$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = \frac{2 x}{\pi}$$
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:
$$\frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}} = - \frac{\left|{x - 2}\right|}{\operatorname{atan}{\left(x - 2 \right)}}$$
- No
$$\frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}} = \frac{\left|{x - 2}\right|}{\operatorname{atan}{\left(x - 2 \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}} = - \frac{\left|{x - 2}\right|}{\operatorname{atan}{\left(x - 2 \right)}}$$
- No
$$\frac{\left|{x + 2}\right|}{\operatorname{atan}{\left(x + 2 \right)}} = \frac{\left|{x - 2}\right|}{\operatorname{atan}{\left(x - 2 \right)}}$$
- No
so, the function
not is
neither even, nor odd