Mister Exam
Graphing y = (x-1/x)*(x+1/x)
The solution
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/ 1\ / 1\
f(x) = |x - -|*|x + -|
\ x/ \ x/
$$f{\left(x \right)} = \left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right)$$
f = (x - 1/x)*(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:
$$\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right) = 0$$
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:
$$\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right) = 0$$
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
$$\left(1 - \frac{1}{x^{2}}\right) \left(x - \frac{1}{x}\right) + \left(1 + \frac{1}{x^{2}}\right) \left(x + \frac{1}{x}\right) = 0$$
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
$$\left(1 - \frac{1}{x^{2}}\right) \left(x - \frac{1}{x}\right) + \left(1 + \frac{1}{x^{2}}\right) \left(x + \frac{1}{x}\right) = 0$$
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
$$2 \left(\left(1 - \frac{1}{x^{2}}\right) \left(1 + \frac{1}{x^{2}}\right) + \frac{x - \frac{1}{x}}{x^{3}} - \frac{x + \frac{1}{x}}{x^{3}}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \sqrt[4]{3}$$
$$x_{2} = \sqrt[4]{3}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(2 \left(\left(1 - \frac{1}{x^{2}}\right) \left(1 + \frac{1}{x^{2}}\right) + \frac{x - \frac{1}{x}}{x^{3}} - \frac{x + \frac{1}{x}}{x^{3}}\right)\right) = -\infty$$
$$\lim_{x \to 0^+}\left(2 \left(\left(1 - \frac{1}{x^{2}}\right) \left(1 + \frac{1}{x^{2}}\right) + \frac{x - \frac{1}{x}}{x^{3}} - \frac{x + \frac{1}{x}}{x^{3}}\right)\right) = -\infty$$
- limits are equal, then skip the corresponding point
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
$$\left(-\infty, - \sqrt[4]{3}\right] \cup \left[\sqrt[4]{3}, \infty\right)$$
Convex at the intervals
$$\left[- \sqrt[4]{3}, \sqrt[4]{3}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(\left(1 - \frac{1}{x^{2}}\right) \left(1 + \frac{1}{x^{2}}\right) + \frac{x - \frac{1}{x}}{x^{3}} - \frac{x + \frac{1}{x}}{x^{3}}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \sqrt[4]{3}$$
$$x_{2} = \sqrt[4]{3}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(2 \left(\left(1 - \frac{1}{x^{2}}\right) \left(1 + \frac{1}{x^{2}}\right) + \frac{x - \frac{1}{x}}{x^{3}} - \frac{x + \frac{1}{x}}{x^{3}}\right)\right) = -\infty$$
$$\lim_{x \to 0^+}\left(2 \left(\left(1 - \frac{1}{x^{2}}\right) \left(1 + \frac{1}{x^{2}}\right) + \frac{x - \frac{1}{x}}{x^{3}} - \frac{x + \frac{1}{x}}{x^{3}}\right)\right) = -\infty$$
- limits are equal, then skip the corresponding point
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
$$\left(-\infty, - \sqrt[4]{3}\right] \cup \left[\sqrt[4]{3}, \infty\right)$$
Convex at the intervals
$$\left[- \sqrt[4]{3}, \sqrt[4]{3}\right]$$
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
$$\lim_{x \to -\infty}\left(\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right)\right) = \infty$$
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 (x - 1/x)*(x + 1/x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right)}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right)}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right)}{x}\right) = -\infty$$
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right)}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right) = \left(- x - \frac{1}{x}\right) \left(- x + \frac{1}{x}\right)$$
- No
$$\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right) = - \left(- x - \frac{1}{x}\right) \left(- x + \frac{1}{x}\right)$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right) = \left(- x - \frac{1}{x}\right) \left(- x + \frac{1}{x}\right)$$
- No
$$\left(x - \frac{1}{x}\right) \left(x + \frac{1}{x}\right) = - \left(- x - \frac{1}{x}\right) \left(- x + \frac{1}{x}\right)$$
- No
so, the function
not is
neither even, nor odd