Mister Exam
Graphing y = -sqrt(2)*sqrt(-1/(-1-x))/2
The solution
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\/ -1 - x
f(x) = -------------------
2
$$f{\left(x \right)} = \frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2}$$
f = ((-sqrt(2))*sqrt(-1/(-x - 1)))/2
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:
$$\frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2} = 0$$
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:
$$\frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2} = 0$$
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
$$\frac{\sqrt{2} \sqrt{- \frac{1}{- x - 1}} \left(x + 1\right)}{4 \left(- x - 1\right)^{2}} = 0$$
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
$$\frac{\sqrt{2} \sqrt{- \frac{1}{- x - 1}} \left(x + 1\right)}{4 \left(- x - 1\right)^{2}} = 0$$
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
$$- \frac{3 \sqrt{2} \sqrt{\frac{1}{x + 1}}}{8 \left(x + 1\right)^{2}} = 0$$
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
$$- \frac{3 \sqrt{2} \sqrt{\frac{1}{x + 1}}}{8 \left(x + 1\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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
$$\lim_{x \to -\infty}\left(\frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of ((-sqrt(2))*sqrt(-1/(-1 - x)))/2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(- \frac{\sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(- \frac{\sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(- \frac{\sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(- \frac{\sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2 x}\right) = 0$$
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:
$$\frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2} = - \frac{\sqrt{2} \sqrt{- \frac{1}{x - 1}}}{2}$$
- No
$$\frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2} = \frac{\sqrt{2} \sqrt{- \frac{1}{x - 1}}}{2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2} = - \frac{\sqrt{2} \sqrt{- \frac{1}{x - 1}}}{2}$$
- No
$$\frac{- \sqrt{2} \sqrt{- \frac{1}{- x - 1}}}{2} = \frac{\sqrt{2} \sqrt{- \frac{1}{x - 1}}}{2}$$
- No
so, the function
not is
neither even, nor odd