Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • (x^2+5)/(x-3)
  • x^2-|4x+5|
  • x^2+5x+4
  • x^3+6x^2+9x

  • Identical expressions

  • one /(four *sqrt(five))*log(two +sqrt(five *tanh(x)))/(two -sqrt(five *tanh(x)))
  • 1 divide by (4 multiply by square root of (5)) multiply by logarithm of (2 plus square root of (5 multiply by hyperbolic tangent of gent of (x))) divide by (2 minus square root of (5 multiply by hyperbolic tangent of gent of (x)))
  • one divide by (four multiply by square root of (five)) multiply by logarithm of (two plus square root of (five multiply by hyperbolic tangent of gent of (x))) divide by (two minus square root of (five multiply by hyperbolic tangent of gent of (x)))
  • 1/(4*√(5))*log(2+√(5*tanh(x)))/(2-√(5*tanh(x)))
  • 1/(4sqrt(5))log(2+sqrt(5tanh(x)))/(2-sqrt(5tanh(x)))
  • 1/4sqrt5log2+sqrt5tanhx/2-sqrt5tanhx
  • 1 divide by (4*sqrt(5))*log(2+sqrt(5*tanh(x))) divide by (2-sqrt(5*tanh(x)))

  • Similar expressions

  • 1/(4*sqrt(5))*log(2+sqrt(5*tanh(x)))/(2+sqrt(5*tanh(x)))
  • 1/(4*sqrt(5))*log(2-sqrt(5*tanh(x)))/(2-sqrt(5*tanh(x)))
Plotting a function graph/ 1/(4*sqrt(5))*log(2+sqrt(5*tanh(x)))/(2-sqrt(5*tanh(x)))

Graphing y = 1/(4*sqrt(5))*log(2+sqrt(5*tanh(x)))/(2-sqrt(5*tanh(x)))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       /   /      ___________\\
       |log\2 + \/ 5*tanh(x) /|
       |----------------------|
       |           ___        |
       \       4*\/ 5         /
f(x) = ------------------------
                ___________    
          2 - \/ 5*tanh(x)
$$f{\left(x \right)} = \frac{\frac{1}{4 \sqrt{5}} \log{\left(\sqrt{5 \tanh{\left(x \right)}} + 2 \right)}}{2 - \sqrt{5 \tanh{\left(x \right)}}}$$ 
f = (log(sqrt(5*tanh(x)) + 2)/((4*sqrt(5))))/(2 - sqrt(5*tanh(x)))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 1.09861228866811$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (log(2 + sqrt(5*tanh(x)))/((4*sqrt(5))))/(2 - sqrt(5*tanh(x))).
$$\frac{\frac{1}{4 \sqrt{5}} \log{\left(\sqrt{5 \tanh{\left(0 \right)}} + 2 \right)}}{2 - \sqrt{5 \tanh{\left(0 \right)}}}$$
The result:
$$f{\left(0 \right)} = \frac{\sqrt{5} \log{\left(2 \right)}}{40}$$
The point:
(0, sqrt(5)*log(2)/40)
Vertical asymptotes
Have:
$$x_{1} = 1.09861228866811$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{4 \sqrt{5}} \log{\left(\sqrt{5 \tanh{\left(x \right)}} + 2 \right)}}{2 - \sqrt{5 \tanh{\left(x \right)}}}\right) = - \frac{\sqrt{5} \log{\left(2 + \sqrt{5} i \right)}}{-40 + 20 \sqrt{5} i}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = - \frac{\sqrt{5} \log{\left(2 + \sqrt{5} i \right)}}{-40 + 20 \sqrt{5} i}$$
$$\lim_{x \to \infty}\left(\frac{\frac{1}{4 \sqrt{5}} \log{\left(\sqrt{5 \tanh{\left(x \right)}} + 2 \right)}}{2 - \sqrt{5 \tanh{\left(x \right)}}}\right) = - \frac{\sqrt{5} \log{\left(2 + \sqrt{5} \right)}}{-40 + 20 \sqrt{5}}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = - \frac{\sqrt{5} \log{\left(2 + \sqrt{5} \right)}}{-40 + 20 \sqrt{5}}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (log(2 + sqrt(5*tanh(x)))/((4*sqrt(5))))/(2 - sqrt(5*tanh(x))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{\sqrt{5}}{20} \log{\left(\sqrt{5 \tanh{\left(x \right)}} + 2 \right)}}{x \left(2 - \sqrt{5 \tanh{\left(x \right)}}\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{\sqrt{5}}{20} \log{\left(\sqrt{5 \tanh{\left(x \right)}} + 2 \right)}}{x \left(2 - \sqrt{5 \tanh{\left(x \right)}}\right)}\right) = 0$$
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:
$$\frac{\frac{1}{4 \sqrt{5}} \log{\left(\sqrt{5 \tanh{\left(x \right)}} + 2 \right)}}{2 - \sqrt{5 \tanh{\left(x \right)}}} = \frac{\sqrt{5} \log{\left(\sqrt{5} \sqrt{- \tanh{\left(x \right)}} + 2 \right)}}{20 \left(- \sqrt{5} \sqrt{- \tanh{\left(x \right)}} + 2\right)}$$
- No
$$\frac{\frac{1}{4 \sqrt{5}} \log{\left(\sqrt{5 \tanh{\left(x \right)}} + 2 \right)}}{2 - \sqrt{5 \tanh{\left(x \right)}}} = - \frac{\sqrt{5} \log{\left(\sqrt{5} \sqrt{- \tanh{\left(x \right)}} + 2 \right)}}{20 \left(- \sqrt{5} \sqrt{- \tanh{\left(x \right)}} + 2\right)}$$
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator