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Plotting a function graph/ sign(sin(/x))

Graphing y = sign(sin(/x))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = sign(sin(x))
f(x)=sign⁡(sin⁡(x))f{\left(x \right)} = \operatorname{sign}{\left(\sin{\left(x \right)} \right)} 
f = sign(sin(x))
The graph of the function
02468-8-6-4-2-10102-2
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:
sign⁡(sin⁡(x))=0\operatorname{sign}{\left(\sin{\left(x \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=0x_{1} = 0
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sign(sin(x)).
sign⁡(sin⁡(0))\operatorname{sign}{\left(\sin{\left(0 \right)} \right)}
The result:
f(0)=sign⁡(sin⁡(0))f{\left(0 \right)} = \operatorname{sign}{\left(\sin{\left(0 \right)} \right)}
The point:
(0, sign(sin(0)))
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
2cos⁡(x)δ(sin⁡(x))=02 \cos{\left(x \right)} \delta\left(\sin{\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
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2(−sin⁡(x)δ(sin⁡(x))+cos⁡2(x)δ(1)(sin⁡(x)))=02 \left(- \sin{\left(x \right)} \delta\left(\sin{\left(x \right)}\right) + \cos^{2}{\left(x \right)} \delta^{\left( 1 \right)}\left( \sin{\left(x \right)} \right)\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→−∞sign⁡(sin⁡(x))=sign⁡(⟨−1,1⟩)\lim_{x \to -\infty} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} = \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=sign⁡(⟨−1,1⟩)y = \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
lim⁡x→∞sign⁡(sin⁡(x))=sign⁡(⟨−1,1⟩)\lim_{x \to \infty} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} = \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=sign⁡(⟨−1,1⟩)y = \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sign(sin(x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sign⁡(sin⁡(x))x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(sign⁡(sin⁡(x))x)=0\lim_{x \to \infty}\left(\frac{\operatorname{sign}{\left(\sin{\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:
sign⁡(sin⁡(x))=−sign⁡(sin⁡(x))\operatorname{sign}{\left(\sin{\left(x \right)} \right)} = - \operatorname{sign}{\left(\sin{\left(x \right)} \right)}
- No
sign⁡(sin⁡(x))=sign⁡(sin⁡(x))\operatorname{sign}{\left(\sin{\left(x \right)} \right)} = \operatorname{sign}{\left(\sin{\left(x \right)} \right)}
- Yes
so, the function
is
odd
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