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Plotting a function graph/ tanhx

Graphing y = tanhx

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The graph:

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Intersection points:

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Piecewise:

The solution

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f(x) = tanh(x)
f(x)=tanh⁡(x)f{\left(x \right)} = \tanh{\left(x \right)} 
f = tanh(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:
tanh⁡(x)=0\tanh{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
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 tanh(x).
tanh⁡(0)\tanh{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 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
1−tanh⁡2(x)=01 - \tanh^{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
2(tanh⁡2(x)−1)tanh⁡(x)=02 \left(\tanh^{2}{\left(x \right)} - 1\right) \tanh{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,0]\left(-\infty, 0\right]
Convex at the intervals
[0,∞)\left[0, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞tanh⁡(x)=−1\lim_{x \to -\infty} \tanh{\left(x \right)} = -1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=−1y = -1
lim⁡x→∞tanh⁡(x)=1\lim_{x \to \infty} \tanh{\left(x \right)} = 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=1y = 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tanh(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(tanh⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\tanh{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(tanh⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\tanh{\left(x \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:
tanh⁡(x)=−tanh⁡(x)\tanh{\left(x \right)} = - \tanh{\left(x \right)}
- No
tanh⁡(x)=tanh⁡(x)\tanh{\left(x \right)} = \tanh{\left(x \right)}
- Yes
so, the function
is
odd
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