Mister Exam
Graphing y = sign(thx)
The solution
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f(x) = sign(tanh(x))
$$f{\left(x \right)} = \operatorname{sign}{\left(\tanh{\left(x \right)} \right)}$$
f = sign(tanh(x))
The graph of the function
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:
$$\operatorname{sign}{\left(\tanh{\left(x \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
so we need to solve the equation:
$$\operatorname{sign}{\left(\tanh{\left(x \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 0$$
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
$$\left(2 - 2 \tanh^{2}{\left(x \right)}\right) \delta\left(\tanh{\left(x \right)}\right) = 0$$
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
$$\left(2 - 2 \tanh^{2}{\left(x \right)}\right) \delta\left(\tanh{\left(x \right)}\right) = 0$$
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
$$2 \left(\left(\tanh^{2}{\left(x \right)} - 1\right) \delta^{\left( 1 \right)}\left( \tanh{\left(x \right)} \right) + 2 \tanh{\left(x \right)} \delta\left(\tanh{\left(x \right)}\right)\right) \left(\tanh^{2}{\left(x \right)} - 1\right) = 0$$
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
$$2 \left(\left(\tanh^{2}{\left(x \right)} - 1\right) \delta^{\left( 1 \right)}\left( \tanh{\left(x \right)} \right) + 2 \tanh{\left(x \right)} \delta\left(\tanh{\left(x \right)}\right)\right) \left(\tanh^{2}{\left(x \right)} - 1\right) = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \operatorname{sign}{\left(\tanh{\left(x \right)} \right)} = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty} \operatorname{sign}{\left(\tanh{\left(x \right)} \right)} = 1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 1$$
$$\lim_{x \to -\infty} \operatorname{sign}{\left(\tanh{\left(x \right)} \right)} = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty} \operatorname{sign}{\left(\tanh{\left(x \right)} \right)} = 1$$
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 sign(tanh(x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{sign}{\left(\tanh{\left(x \right)} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{sign}{\left(\tanh{\left(x \right)} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{sign}{\left(\tanh{\left(x \right)} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{sign}{\left(\tanh{\left(x \right)} \right)}}{x}\right) = 0$$
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:
$$\operatorname{sign}{\left(\tanh{\left(x \right)} \right)} = - \operatorname{sign}{\left(\tanh{\left(x \right)} \right)}$$
- No
$$\operatorname{sign}{\left(\tanh{\left(x \right)} \right)} = \operatorname{sign}{\left(\tanh{\left(x \right)} \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\operatorname{sign}{\left(\tanh{\left(x \right)} \right)} = - \operatorname{sign}{\left(\tanh{\left(x \right)} \right)}$$
- No
$$\operatorname{sign}{\left(\tanh{\left(x \right)} \right)} = \operatorname{sign}{\left(\tanh{\left(x \right)} \right)}$$
- Yes
so, the function
is
odd