Mister Exam

Other calculators

Plotting a function graph/ arсsin(x-11)/((x-12)(x-10))

Graphing y = arсsin(x-11)/((x-12)(x-10))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          asin(x - 11)  
f(x) = -----------------
       (x - 12)*(x - 10)
f(x)=asin⁡(x−11)(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:
x1=10x_{1} = 10
x2=12x_{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:
asin⁡(x−11)(x−12)(x−10)=0\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
x1=11x_{1} = 11
Numerical solution
x1=11x_{1} = 11
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to asin(x - 11)/(((x - 12)*(x - 10))).
asin⁡(−11)(−10)(−1)12\frac{\operatorname{asin}{\left(-11 \right)}}{\left(-10\right) \left(-1\right) 12}
The result:
f(0)=−asin⁡(11)120f{\left(0 \right)} = - \frac{\operatorname{asin}{\left(11 \right)}}{120}
The point:
(0, -asin(11)/120)
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
(22−2x)asin⁡(x−11)(x−12)2(x−10)2+1x−121x−101−(x−11)2=0\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
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
2((x−11)(1x−10+1x−12)+x−11x−10−1+x−11x−12)asin⁡(x−11)(x−12)(x−10)−4(x−11)1−(x−11)2(x−12)(x−10)+x−11(1−(x−11)2)32(x−12)(x−10)=0\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:
x1=10x_{1} = 10
x2=12x_{2} = 12
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→−∞(asin⁡(x−11)(x−12)(x−10))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→∞(asin⁡(x−11)(x−12)(x−10))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
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(1(x−12)(x−10)asin⁡(x−11)x)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=xlim⁡x→∞(1(x−12)(x−10)asin⁡(x−11)x)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:
asin⁡(x−11)(x−12)(x−10)=−asin⁡(x+11)(−x−12)(−x−10)\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
asin⁡(x−11)(x−12)(x−10)=asin⁡(x+11)(−x−12)(−x−10)\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
    © Mister Exam – Calculator