Mister Exam
Graphing y = sqrt((3x-2)/(x+1)-2+2*arccos((5-2x)\4))
The solution
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/ 3*x - 2 /5 - 2*x\
f(x) = / ------- - 2 + 2*acos|-------|
\/ x + 1 \ 4 /
$$f{\left(x \right)} = \sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}}$$
f = sqrt(-2 + (3*x - 2)/(x + 1) + 2*acos((5 - 2*x)/4))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -1$$
$$x_{1} = -1$$
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:
$$\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$\frac{\frac{3}{2 \left(x + 1\right)} - \frac{3 x - 2}{2 \left(x + 1\right)^{2}} + \frac{1}{2 \sqrt{1 - \frac{\left(5 - 2 x\right)^{2}}{16}}}}{\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}}} = 0$$
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
$$\frac{\frac{3}{2 \left(x + 1\right)} - \frac{3 x - 2}{2 \left(x + 1\right)^{2}} + \frac{1}{2 \sqrt{1 - \frac{\left(5 - 2 x\right)^{2}}{16}}}}{\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Vertical asymptotes
Have:
$$x_{1} = -1$$
$$x_{1} = -1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}} = \infty \operatorname{sign}{\left(\sqrt{i} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \infty \operatorname{sign}{\left(\sqrt{i} \right)}$$
$$\lim_{x \to \infty} \sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}} = \infty \operatorname{sign}{\left(\sqrt{- i} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \infty \operatorname{sign}{\left(\sqrt{- i} \right)}$$
$$\lim_{x \to -\infty} \sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}} = \infty \operatorname{sign}{\left(\sqrt{i} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \infty \operatorname{sign}{\left(\sqrt{i} \right)}$$
$$\lim_{x \to \infty} \sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}} = \infty \operatorname{sign}{\left(\sqrt{- i} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \infty \operatorname{sign}{\left(\sqrt{- i} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt((3*x - 2)/(x + 1) - 2 + 2*acos((5 - 2*x)/4)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}}}{x}\right) = 0$$
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:
$$\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}} = \sqrt{2 \operatorname{acos}{\left(\frac{x}{2} + \frac{5}{4} \right)} - 2 + \frac{- 3 x - 2}{1 - x}}$$
- No
$$\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}} = - \sqrt{2 \operatorname{acos}{\left(\frac{x}{2} + \frac{5}{4} \right)} - 2 + \frac{- 3 x - 2}{1 - x}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}} = \sqrt{2 \operatorname{acos}{\left(\frac{x}{2} + \frac{5}{4} \right)} - 2 + \frac{- 3 x - 2}{1 - x}}$$
- No
$$\sqrt{\left(-2 + \frac{3 x - 2}{x + 1}\right) + 2 \operatorname{acos}{\left(\frac{5 - 2 x}{4} \right)}} = - \sqrt{2 \operatorname{acos}{\left(\frac{x}{2} + \frac{5}{4} \right)} - 2 + \frac{- 3 x - 2}{1 - x}}$$
- No
so, the function
not is
neither even, nor odd