Mister Exam
Graphing y = asin(x-1/x)
The solution
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/ 1\
f(x) = asin|x - -|
\ x/
$$f{\left(x \right)} = \operatorname{asin}{\left(x - \frac{1}{x} \right)}$$
f = asin(x - 1/x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\operatorname{asin}{\left(x - \frac{1}{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
$$x_{2} = 1$$
Numerical solution
$$x_{1} = -1$$
$$x_{2} = 1$$
so we need to solve the equation:
$$\operatorname{asin}{\left(x - \frac{1}{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -1$$
$$x_{2} = 1$$
Numerical solution
$$x_{1} = -1$$
$$x_{2} = 1$$
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
$$\frac{1 + \frac{1}{x^{2}}}{\sqrt{1 - \left(x - \frac{1}{x}\right)^{2}}} = 0$$
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
$$\frac{1 + \frac{1}{x^{2}}}{\sqrt{1 - \left(x - \frac{1}{x}\right)^{2}}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \operatorname{asin}{\left(x - \frac{1}{x} \right)} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \operatorname{asin}{\left(x - \frac{1}{x} \right)} = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \operatorname{asin}{\left(x - \frac{1}{x} \right)} = \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \operatorname{asin}{\left(x - \frac{1}{x} \right)} = - \infty i$$
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 asin(x - 1/x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x - \frac{1}{x} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x - \frac{1}{x} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x - \frac{1}{x} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x - \frac{1}{x} \right)}}{x}\right)$$
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:
$$\operatorname{asin}{\left(x - \frac{1}{x} \right)} = - \operatorname{asin}{\left(x - \frac{1}{x} \right)}$$
- No
$$\operatorname{asin}{\left(x - \frac{1}{x} \right)} = \operatorname{asin}{\left(x - \frac{1}{x} \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\operatorname{asin}{\left(x - \frac{1}{x} \right)} = - \operatorname{asin}{\left(x - \frac{1}{x} \right)}$$
- No
$$\operatorname{asin}{\left(x - \frac{1}{x} \right)} = \operatorname{asin}{\left(x - \frac{1}{x} \right)}$$
- Yes
so, the function
is
odd