Mister Exam

Other calculators

arcsin(x+1)-1
Plotting a function graph/ arcsin(x+1)-1

Graphing y = arcsin(x+1)-1

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = asin(x + 1) - 1
f(x)=asin⁡(x+1)−1f{\left(x \right)} = \operatorname{asin}{\left(x + 1 \right)} - 1 
f = asin(x + 1) - 1*1
The graph of the function
02468-8-6-4-2-10102.5-2.5
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+1)−1=0\operatorname{asin}{\left(x + 1 \right)} - 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−1+sin⁡(1)x_{1} = -1 + \sin{\left(1 \right)}
Numerical solution
x1=−0.158529015192103x_{1} = -0.158529015192103
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) - 1*1.
(−1)1+asin⁡(0+1)\left(-1\right) 1 + \operatorname{asin}{\left(0 + 1 \right)}
The result:
f(0)=−1+π2f{\left(0 \right)} = -1 + \frac{\pi}{2}
The point:
(0, -1 + pi/2)
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
1−(x+1)2+1=0\frac{1}{\sqrt{- \left(x + 1\right)^{2} + 1}} = 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
x+1(−(x+1)2+1)32=0\frac{x + 1}{\left(- \left(x + 1\right)^{2} + 1\right)^{\frac{3}{2}}} = 0
Solve this equation
The roots of this equation
x1=−1x_{1} = -1

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
[−1,∞)\left[-1, \infty\right)
Convex at the intervals
(−∞,−1]\left(-\infty, -1\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⁡(x+1)−1)=∞i\lim_{x \to -\infty}\left(\operatorname{asin}{\left(x + 1 \right)} - 1\right) = \infty i
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞(asin⁡(x+1)−1)=−∞i\lim_{x \to \infty}\left(\operatorname{asin}{\left(x + 1 \right)} - 1\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(x + 1) - 1*1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(asin⁡(x+1)−1x)=0\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x + 1 \right)} - 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(asin⁡(x+1)−1x)=0\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x + 1 \right)} - 1}{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⁡(x+1)−1=−asin⁡(x−1)−1\operatorname{asin}{\left(x + 1 \right)} - 1 = - \operatorname{asin}{\left(x - 1 \right)} - 1
- No
asin⁡(x+1)−1=asin⁡(x−1)+1\operatorname{asin}{\left(x + 1 \right)} - 1 = \operatorname{asin}{\left(x - 1 \right)} + 1
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = arcsin(x+1)-1
    © Mister Exam – Calculator