Graphing y = arcsin(x-1)/lgx

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       asin(x - 1)
f(x) = -----------
          log(x)

f{\left(x \right)} = \frac{\operatorname{asin}{\left(x - 1 \right)}}{\log{\left(x \right)}}

f = asin(x - 1)/log(x)

The graph of the function

The domain of the function

The points at which the function is not precisely defined:

x_{1} = 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:

\frac{\operatorname{asin}{\left(x - 1 \right)}}{\log{\left(x \right)}} = 0

Solve this equation
The points of intersection with the axis X:

Numerical solution

x_{1} = 0

The points of intersection with the Y axis coordinate

The graph crosses Y axis when x equals 0:
substitute x = 0 to asin(x - 1)/log(x).

\frac{\operatorname{asin}{\left(-1 \right)}}{\log{\left(0 \right)}}

The result:

f{\left(0 \right)} = 0

The point:

(0, 0)

Extrema of the function

In order to find the extrema, we need to solve the equation

\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:

\frac{d}{d x} f{\left(x \right)} =

\frac{1}{\sqrt{1 - \left(x - 1\right)^{2}} \log{\left(x \right)}} - \frac{\operatorname{asin}{\left(x - 1 \right)}}{x \log{\left(x \right)}^{2}} = 0

Solve this equation
Solutions are not found,
function may have no extrema

Vertical asymptotes

Have:

x_{1} = 1

Horizontal asymptotes

Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo

True

Let's take the limit
so,
equation of the horizontal asymptote on the left:

y = \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x - 1 \right)}}{\log{\left(x \right)}}\right)

True

Let's take the limit
so,
equation of the horizontal asymptote on the right:

y = \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x - 1 \right)}}{\log{\left(x \right)}}\right)

Inclined asymptotes

Inclined asymptote can be found by calculating the limit of asin(x - 1)/log(x), divided by x at x->+oo and x ->-oo

True

Let's take the limit
so,
inclined asymptote equation on the left:

y = x \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x - 1 \right)}}{x \log{\left(x \right)}}\right)

True

Let's take the limit
so,
inclined asymptote equation on the right:

y = x \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x - 1 \right)}}{x \log{\left(x \right)}}\right)

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{\operatorname{asin}{\left(x - 1 \right)}}{\log{\left(x \right)}} = - \frac{\operatorname{asin}{\left(x + 1 \right)}}{\log{\left(- x \right)}}

- No

\frac{\operatorname{asin}{\left(x - 1 \right)}}{\log{\left(x \right)}} = \frac{\operatorname{asin}{\left(x + 1 \right)}}{\log{\left(- x \right)}}

- No
so, the function
not is
neither even, nor odd