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  • Graphing y =:
  • x^3-3x^2+2x+1
  • (x^2-1)/(x-1)
  • 3x^2-2x
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  • Identical expressions

  • asin(log(five *x))
  • arc sinus of e of ( logarithm of (5 multiply by x))
  • arc sinus of e of ( logarithm of (five multiply by x))
  • asin(log(5x))
  • asinlog5x

  • Similar expressions

  • arcsin(log(5*x))
Plotting a function graph/ asin(log(5*x))

Graphing y = asin(log(5*x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = asin(log(5*x))
f(x)=asin⁡(log⁡(5x))f{\left(x \right)} = \operatorname{asin}{\left(\log{\left(5 x \right)} \right)} 
f = asin(log(5*x))
The graph of the function
02468-8-6-4-2-10101-1
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:
asin⁡(log⁡(5x))=0\operatorname{asin}{\left(\log{\left(5 x \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=15x_{1} = \frac{1}{5}
Numerical solution
x1=0.2x_{1} = 0.2
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to asin(log(5*x)).
asin⁡(log⁡(0⋅5))\operatorname{asin}{\left(\log{\left(0 \cdot 5 \right)} \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
1x1−log⁡(5x)2=0\frac{1}{x \sqrt{1 - \log{\left(5 x \right)}^{2}}} = 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+log⁡(5x)1−log⁡(5x)2x21−log⁡(5x)2=0\frac{-1 + \frac{\log{\left(5 x \right)}}{1 - \log{\left(5 x \right)}^{2}}}{x^{2} \sqrt{1 - \log{\left(5 x \right)}^{2}}} = 0
Solve this equation
The roots of this equation
x1=e−12+525x_{1} = \frac{e^{- \frac{1}{2} + \frac{\sqrt{5}}{2}}}{5}
x2=15e12+52x_{2} = \frac{1}{5 e^{\frac{1}{2} + \frac{\sqrt{5}}{2}}}

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
[e−12+525,∞)\left[\frac{e^{- \frac{1}{2} + \frac{\sqrt{5}}{2}}}{5}, \infty\right)
Convex at the intervals
(−∞,e−12+525]\left(-\infty, \frac{e^{- \frac{1}{2} + \frac{\sqrt{5}}{2}}}{5}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞asin⁡(log⁡(5x))=−∞i\lim_{x \to -\infty} \operatorname{asin}{\left(\log{\left(5 x \right)} \right)} = - \infty i
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞asin⁡(log⁡(5x))=−∞i\lim_{x \to \infty} \operatorname{asin}{\left(\log{\left(5 x \right)} \right)} = - \infty i
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of asin(log(5*x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(asin⁡(log⁡(5x))x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\log{\left(5 x \right)} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(asin⁡(log⁡(5x))x)=0\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(\log{\left(5 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:
asin⁡(log⁡(5x))=asin⁡(log⁡(−5x))\operatorname{asin}{\left(\log{\left(5 x \right)} \right)} = \operatorname{asin}{\left(\log{\left(- 5 x \right)} \right)}
- No
asin⁡(log⁡(5x))=−asin⁡(log⁡(−5x))\operatorname{asin}{\left(\log{\left(5 x \right)} \right)} = - \operatorname{asin}{\left(\log{\left(- 5 x \right)} \right)}
- No
so, the function
not is
neither even, nor odd
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