Mister Exam
Graphing y = arcsin(x-1)/lgx
The solution
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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:
$$x_{1} = 1$$
$$x_{1} = 1$$
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
$$x_{1} = 0$$
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
$$x_{1} = 0$$
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:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{1}{\sqrt{1 - \left(x - 1\right)^{2}} \log{\left(x \right)}} - \frac{\operatorname{asin}{\left(x - 1 \right)}}{x \log{\left(x \right)}^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{1}{\sqrt{1 - \left(x - 1\right)^{2}} \log{\left(x \right)}} - \frac{\operatorname{asin}{\left(x - 1 \right)}}{x \log{\left(x \right)}^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Vertical asymptotes
Have:
$$x_{1} = 1$$
$$x_{1} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x - 1 \right)}}{\log{\left(x \right)}}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x - 1 \right)}}{\log{\left(x \right)}}\right)$$
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
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x - 1 \right)}}{x \log{\left(x \right)}}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x - 1 \right)}}{x \log{\left(x \right)}}\right)$$
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
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