Mister Exam
Graphing y = 2(ln(x/(x+1)))-1
The solution
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/ x \
f(x) = 2*log|-----| - 1
\x + 1/
$$f{\left(x \right)} = 2 \log{\left(\frac{x}{x + 1} \right)} - 1$$
f = 2*log(x/(x + 1)) - 1
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:
$$2 \log{\left(\frac{x}{x + 1} \right)} - 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{e^{\frac{1}{2}}}{1 - e^{\frac{1}{2}}}$$
Numerical solution
$$x_{1} = -2.5414940825368$$
so we need to solve the equation:
$$2 \log{\left(\frac{x}{x + 1} \right)} - 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{e^{\frac{1}{2}}}{1 - e^{\frac{1}{2}}}$$
Numerical solution
$$x_{1} = -2.5414940825368$$
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
$$\frac{2 \left(x + 1\right) \left(- \frac{x}{\left(x + 1\right)^{2}} + \frac{1}{x + 1}\right)}{x} = 0$$
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
$$\frac{2 \left(x + 1\right) \left(- \frac{x}{\left(x + 1\right)^{2}} + \frac{1}{x + 1}\right)}{x} = 0$$
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
$$\frac{2 \left(\frac{x}{x + 1} - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x}\right)}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{1}{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$\lim_{x \to -1^-}\left(\frac{2 \left(\frac{x}{x + 1} - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x}\right)}{x}\right) = \infty$$
$$\lim_{x \to -1^+}\left(\frac{2 \left(\frac{x}{x + 1} - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x}\right)}{x}\right) = \infty$$
- 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
$$\left(-\infty, - \frac{1}{2}\right]$$
Convex at the intervals
$$\left[- \frac{1}{2}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(\frac{x}{x + 1} - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x}\right)}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{1}{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$\lim_{x \to -1^-}\left(\frac{2 \left(\frac{x}{x + 1} - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x}\right)}{x}\right) = \infty$$
$$\lim_{x \to -1^+}\left(\frac{2 \left(\frac{x}{x + 1} - 1\right) \left(\frac{1}{x + 1} + \frac{1}{x}\right)}{x}\right) = \infty$$
- 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
$$\left(-\infty, - \frac{1}{2}\right]$$
Convex at the intervals
$$\left[- \frac{1}{2}, \infty\right)$$
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
$$\lim_{x \to -\infty}\left(2 \log{\left(\frac{x}{x + 1} \right)} - 1\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(2 \log{\left(\frac{x}{x + 1} \right)} - 1\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -1$$
$$\lim_{x \to -\infty}\left(2 \log{\left(\frac{x}{x + 1} \right)} - 1\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -1$$
$$\lim_{x \to \infty}\left(2 \log{\left(\frac{x}{x + 1} \right)} - 1\right) = -1$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -1$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*log(x/(x + 1)) - 1, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{2 \log{\left(\frac{x}{x + 1} \right)} - 1}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 \log{\left(\frac{x}{x + 1} \right)} - 1}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{2 \log{\left(\frac{x}{x + 1} \right)} - 1}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 \log{\left(\frac{x}{x + 1} \right)} - 1}{x}\right) = 0$$
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:
$$2 \log{\left(\frac{x}{x + 1} \right)} - 1 = 2 \log{\left(- \frac{x}{1 - x} \right)} - 1$$
- No
$$2 \log{\left(\frac{x}{x + 1} \right)} - 1 = 1 - 2 \log{\left(- \frac{x}{1 - x} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$2 \log{\left(\frac{x}{x + 1} \right)} - 1 = 2 \log{\left(- \frac{x}{1 - x} \right)} - 1$$
- No
$$2 \log{\left(\frac{x}{x + 1} \right)} - 1 = 1 - 2 \log{\left(- \frac{x}{1 - x} \right)}$$
- No
so, the function
not is
neither even, nor odd