Mister Exam

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  • How to use it?

  • Graphing y =:
  • x^3-12x+24
  • -x^2+5*x+4
  • x^2+4x+2
  • (x^2-4x+1)/(x-4)
  • Identical expressions

  • acos((two *x- one)/ three)
  • arc co sinus of e of ine of ((2 multiply by x minus 1) divide by 3)
  • arc co sinus of e of ine of ((two multiply by x minus one) divide by three)
  • acos((2x-1)/3)
  • acos2x-1/3
  • acos((2*x-1) divide by 3)
  • Similar expressions

  • acos((2*x+1)/3)
  • arccos((2*x-1)/3)

Graphing y = acos((2*x-1)/3)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src]
           /2*x - 1\
f(x) = acos|-------|
           \   3   /
$$f{\left(x \right)} = \operatorname{acos}{\left(\frac{2 x - 1}{3} \right)}$$
f = acos((2*x - 1)/3)
The graph of the function
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:
$$\operatorname{acos}{\left(\frac{2 x - 1}{3} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:

Analytical solution
$$x_{1} = 2$$
Numerical solution
$$x_{1} = 2$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to acos((2*x - 1)/3).
$$\operatorname{acos}{\left(\frac{-1 + 0 \cdot 2}{3} \right)}$$
The result:
$$f{\left(0 \right)} = \operatorname{acos}{\left(- \frac{1}{3} \right)}$$
The point:
(0, acos(-1/3))
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
$$- \frac{2}{3 \sqrt{1 - \frac{\left(2 x - 1\right)^{2}}{9}}} = 0$$
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
$$- \frac{4 \left(2 x - 1\right)}{27 \left(1 - \frac{\left(2 x - 1\right)^{2}}{9}\right)^{\frac{3}{2}}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{1}{2}$$

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
$$\left(-\infty, \frac{1}{2}\right]$$
Convex at the intervals
$$\left[\frac{1}{2}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \operatorname{acos}{\left(\frac{2 x - 1}{3} \right)} = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \operatorname{acos}{\left(\frac{2 x - 1}{3} \right)} = \infty i$$
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 acos((2*x - 1)/3), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acos}{\left(\frac{2 x - 1}{3} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{acos}{\left(\frac{2 x - 1}{3} \right)}}{x}\right) = 0$$
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:
$$\operatorname{acos}{\left(\frac{2 x - 1}{3} \right)} = \operatorname{acos}{\left(- \frac{2 x}{3} - \frac{1}{3} \right)}$$
- No
$$\operatorname{acos}{\left(\frac{2 x - 1}{3} \right)} = - \operatorname{acos}{\left(- \frac{2 x}{3} - \frac{1}{3} \right)}$$
- No
so, the function
not is
neither even, nor odd