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Plotting a function graph/ arccos(sqrt(x))

Graphing y = arccos(sqrt(x))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = acos\\/ x /
f(x)=acos⁡(x)f{\left(x \right)} = \operatorname{acos}{\left(\sqrt{x} \right)} 
f = acos(sqrt(x))
The graph of the function
02468-8-6-4-2-101002
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:
acos⁡(x)=0\operatorname{acos}{\left(\sqrt{x} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=1x_{1} = 1
Numerical solution
x1=1x_{1} = 1
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to acos(sqrt(x)).
acos⁡(0)\operatorname{acos}{\left(\sqrt{0} \right)}
The result:
f(0)=π2f{\left(0 \right)} = \frac{\pi}{2}
The point:
(0, 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
−12x1−x=0- \frac{1}{2 \sqrt{x} \sqrt{1 - x}} = 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
−11−x+1x4x1−x=0\frac{- \frac{1}{1 - x} + \frac{1}{x}}{4 \sqrt{x} \sqrt{1 - x}} = 0
Solve this equation
The roots of this equation
x1=12x_{1} = \frac{1}{2}

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
(−∞,12]\left(-\infty, \frac{1}{2}\right]
Convex at the intervals
[12,∞)\left[\frac{1}{2}, \infty\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of acos(sqrt(x)), 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→−∞(acos⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\operatorname{acos}{\left(\sqrt{x} \right)}}{x}\right)
lim⁡x→∞(acos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\operatorname{acos}{\left(\sqrt{x} \right)}}{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:
acos⁡(x)=acos⁡(−x)\operatorname{acos}{\left(\sqrt{x} \right)} = \operatorname{acos}{\left(\sqrt{- x} \right)}
- No
acos⁡(x)=−acos⁡(−x)\operatorname{acos}{\left(\sqrt{x} \right)} = - \operatorname{acos}{\left(\sqrt{- x} \right)}
- No
so, the function
not is
neither even, nor odd
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