Mister Exam
Graphing y = y=sqrt(12+x-x^2)+1/(sqrt(2|0,5-x|-5))
The solution
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/ 2 1
f(x) = \/ 12 + x - x + -------------------
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\/ 2*|1/2 - x| - 5
$$f{\left(x \right)} = \sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}}$$
f = sqrt(-x^2 + x + 12) + 1/(sqrt(2*|1/2 - x| - 5))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -2$$
$$x_{2} = 3$$
$$x_{1} = -2$$
$$x_{2} = 3$$
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:
$$\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}} = 0$$
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:
$$\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}} = 0$$
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
$$\frac{\frac{1}{2} - x}{\sqrt{- x^{2} + \left(x + 12\right)}} + \frac{\operatorname{sign}{\left(\frac{1}{2} - x \right)}}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5} \left(2 \left|{\frac{1}{2} - x}\right| - 5\right)} = 0$$
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
$$\frac{\frac{1}{2} - x}{\sqrt{- x^{2} + \left(x + 12\right)}} + \frac{\operatorname{sign}{\left(\frac{1}{2} - x \right)}}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5} \left(2 \left|{\frac{1}{2} - x}\right| - 5\right)} = 0$$
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
$$- \frac{\left(2 x - 1\right)^{2}}{4 \left(- x^{2} + x + 12\right)^{\frac{3}{2}}} - \frac{1}{\sqrt{- x^{2} + x + 12}} - \frac{2 \delta\left(x - \frac{1}{2}\right)}{\left(2 \left|{x - \frac{1}{2}}\right| - 5\right)^{\frac{3}{2}}} + \frac{3 \operatorname{sign}^{2}{\left(x - \frac{1}{2} \right)}}{\left(2 \left|{x - \frac{1}{2}}\right| - 5\right)^{\frac{5}{2}}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\left(2 x - 1\right)^{2}}{4 \left(- x^{2} + x + 12\right)^{\frac{3}{2}}} - \frac{1}{\sqrt{- x^{2} + x + 12}} - \frac{2 \delta\left(x - \frac{1}{2}\right)}{\left(2 \left|{x - \frac{1}{2}}\right| - 5\right)^{\frac{3}{2}}} + \frac{3 \operatorname{sign}^{2}{\left(x - \frac{1}{2} \right)}}{\left(2 \left|{x - \frac{1}{2}}\right| - 5\right)^{\frac{5}{2}}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = -2$$
$$x_{2} = 3$$
$$x_{1} = -2$$
$$x_{2} = 3$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}}\right) = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}}\right) = \infty i$$
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 sqrt(12 + x - x^2) + 1/(sqrt(2*|1/2 - x| - 5)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}}}{x}\right) = - i$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - i x$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}}}{x}\right) = i$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = i x$$
$$\lim_{x \to -\infty}\left(\frac{\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}}}{x}\right) = - i$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = - i x$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}}}{x}\right) = i$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = i x$$
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:
$$\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}} = \sqrt{- x^{2} - x + 12} + \frac{1}{\sqrt{2 \left|{x + \frac{1}{2}}\right| - 5}}$$
- No
$$\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}} = - \sqrt{- x^{2} - x + 12} - \frac{1}{\sqrt{2 \left|{x + \frac{1}{2}}\right| - 5}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}} = \sqrt{- x^{2} - x + 12} + \frac{1}{\sqrt{2 \left|{x + \frac{1}{2}}\right| - 5}}$$
- No
$$\sqrt{- x^{2} + \left(x + 12\right)} + \frac{1}{\sqrt{2 \left|{\frac{1}{2} - x}\right| - 5}} = - \sqrt{- x^{2} - x + 12} - \frac{1}{\sqrt{2 \left|{x + \frac{1}{2}}\right| - 5}}$$
- No
so, the function
not is
neither even, nor odd