Mister Exam

Other calculators

Plotting a function graph/ 1/shx

Graphing y = 1/shx

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          1   
f(x) = -------
       sinh(x)
f(x)=1sinh⁡(x)f{\left(x \right)} = \frac{1}{\sinh{\left(x \right)}} 
f = 1/sinh(x)
The graph of the function
02468-8-6-4-2-1010-10001000
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
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:
1sinh⁡(x)=0\frac{1}{\sinh{\left(x \right)}} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 1/sinh(x).
1sinh⁡(0)\frac{1}{\sinh{\left(0 \right)}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
−cosh⁡(x)sinh⁡2(x)=0- \frac{\cosh{\left(x \right)}}{\sinh^{2}{\left(x \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
−1+2cosh⁡2(x)sinh⁡2(x)sinh⁡(x)=0\frac{-1 + \frac{2 \cosh^{2}{\left(x \right)}}{\sinh^{2}{\left(x \right)}}}{\sinh{\left(x \right)}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=0x_{1} = 0
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞1sinh⁡(x)=0\lim_{x \to -\infty} \frac{1}{\sinh{\left(x \right)}} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞1sinh⁡(x)=0\lim_{x \to \infty} \frac{1}{\sinh{\left(x \right)}} = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 1/sinh(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(1xsinh⁡(x))=0\lim_{x \to -\infty}\left(\frac{1}{x \sinh{\left(x \right)}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(1xsinh⁡(x))=0\lim_{x \to \infty}\left(\frac{1}{x \sinh{\left(x \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:
1sinh⁡(x)=−1sinh⁡(x)\frac{1}{\sinh{\left(x \right)}} = - \frac{1}{\sinh{\left(x \right)}}
- No
1sinh⁡(x)=1sinh⁡(x)\frac{1}{\sinh{\left(x \right)}} = \frac{1}{\sinh{\left(x \right)}}
- Yes
so, the function
is
odd
    © Mister Exam – Calculator