Mister Exam
Graphing y = arсsin(x-11)/((x-12)(x-10))
The solution
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asin(x - 11)
f(x) = -----------------
(x - 12)*(x - 10)
$$f{\left(x \right)} = \frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)}$$
f = asin(x - 11)/(((x - 12)*(x - 10)))
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 10$$
$$x_{2} = 12$$
$$x_{1} = 10$$
$$x_{2} = 12$$
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:
$$\frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 11$$
Numerical solution
$$x_{1} = 11$$
so we need to solve the equation:
$$\frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 11$$
Numerical solution
$$x_{1} = 11$$
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
$$\frac{\left(22 - 2 x\right) \operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right)^{2} \left(x - 10\right)^{2}} + \frac{\frac{1}{x - 12} \frac{1}{x - 10}}{\sqrt{1 - \left(x - 11\right)^{2}}} = 0$$
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
$$\frac{\left(22 - 2 x\right) \operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right)^{2} \left(x - 10\right)^{2}} + \frac{\frac{1}{x - 12} \frac{1}{x - 10}}{\sqrt{1 - \left(x - 11\right)^{2}}} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\frac{2 \left(\left(x - 11\right) \left(\frac{1}{x - 10} + \frac{1}{x - 12}\right) + \frac{x - 11}{x - 10} - 1 + \frac{x - 11}{x - 12}\right) \operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)} - \frac{4 \left(x - 11\right)}{\sqrt{1 - \left(x - 11\right)^{2}} \left(x - 12\right) \left(x - 10\right)} + \frac{x - 11}{\left(1 - \left(x - 11\right)^{2}\right)^{\frac{3}{2}}}}{\left(x - 12\right) \left(x - 10\right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\frac{2 \left(\left(x - 11\right) \left(\frac{1}{x - 10} + \frac{1}{x - 12}\right) + \frac{x - 11}{x - 10} - 1 + \frac{x - 11}{x - 12}\right) \operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)} - \frac{4 \left(x - 11\right)}{\sqrt{1 - \left(x - 11\right)^{2}} \left(x - 12\right) \left(x - 10\right)} + \frac{x - 11}{\left(1 - \left(x - 11\right)^{2}\right)^{\frac{3}{2}}}}{\left(x - 12\right) \left(x - 10\right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 10$$
$$x_{2} = 12$$
$$x_{1} = 10$$
$$x_{2} = 12$$
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:
$$y = \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of asin(x - 11)/(((x - 12)*(x - 10))), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{\left(x - 12\right) \left(x - 10\right)} \operatorname{asin}{\left(x - 11 \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{1}{\left(x - 12\right) \left(x - 10\right)} \operatorname{asin}{\left(x - 11 \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{\left(x - 12\right) \left(x - 10\right)} \operatorname{asin}{\left(x - 11 \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{1}{\left(x - 12\right) \left(x - 10\right)} \operatorname{asin}{\left(x - 11 \right)}}{x}\right)$$
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:
$$\frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)} = - \frac{\operatorname{asin}{\left(x + 11 \right)}}{\left(- x - 12\right) \left(- x - 10\right)}$$
- No
$$\frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)} = \frac{\operatorname{asin}{\left(x + 11 \right)}}{\left(- x - 12\right) \left(- x - 10\right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)} = - \frac{\operatorname{asin}{\left(x + 11 \right)}}{\left(- x - 12\right) \left(- x - 10\right)}$$
- No
$$\frac{\operatorname{asin}{\left(x - 11 \right)}}{\left(x - 12\right) \left(x - 10\right)} = \frac{\operatorname{asin}{\left(x + 11 \right)}}{\left(- x - 12\right) \left(- x - 10\right)}$$
- No
so, the function
not is
neither even, nor odd