Mister Exam
Graphing y = ln((2x-1)/(x^2-9))
The solution
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/2*x - 1\
f(x) = log|-------|
| 2 |
\ x - 9/
$$f{\left(x \right)} = \log{\left(\frac{2 x - 1}{x^{2} - 9} \right)}$$
f = log((2*x - 1)/(x^2 - 9))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -3$$
$$x_{2} = 3$$
$$x_{1} = -3$$
$$x_{2} = 3$$
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:
$$\log{\left(\frac{2 x - 1}{x^{2} - 9} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
$$x_{2} = 4$$
Numerical solution
$$x_{1} = -2$$
$$x_{2} = 4$$
so we need to solve the equation:
$$\log{\left(\frac{2 x - 1}{x^{2} - 9} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -2$$
$$x_{2} = 4$$
Numerical solution
$$x_{1} = -2$$
$$x_{2} = 4$$
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
$$\frac{\left(x^{2} - 9\right) \left(- \frac{2 x \left(2 x - 1\right)}{\left(x^{2} - 9\right)^{2}} + \frac{2}{x^{2} - 9}\right)}{2 x - 1} = 0$$
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
$$\frac{\left(x^{2} - 9\right) \left(- \frac{2 x \left(2 x - 1\right)}{\left(x^{2} - 9\right)^{2}} + \frac{2}{x^{2} - 9}\right)}{2 x - 1} = 0$$
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
$$\frac{2 \left(- \frac{2 x \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} - 9} + 6 x - 1}{x^{2} - 9} + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{2 x - 1}\right)}{2 x - 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2} + \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2}$$
$$x_{2} = - \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2} - \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -3$$
$$x_{2} = 3$$
$$\lim_{x \to -3^-}\left(\frac{2 \left(- \frac{2 x \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} - 9} + 6 x - 1}{x^{2} - 9} + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{2 x - 1}\right)}{2 x - 1}\right) = \infty$$
$$\lim_{x \to -3^+}\left(\frac{2 \left(- \frac{2 x \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} - 9} + 6 x - 1}{x^{2} - 9} + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{2 x - 1}\right)}{2 x - 1}\right) = \infty$$
- limits are equal, then skip the corresponding point
$$\lim_{x \to 3^-}\left(\frac{2 \left(- \frac{2 x \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} - 9} + 6 x - 1}{x^{2} - 9} + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{2 x - 1}\right)}{2 x - 1}\right) = \infty$$
$$\lim_{x \to 3^+}\left(\frac{2 \left(- \frac{2 x \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} - 9} + 6 x - 1}{x^{2} - 9} + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{2 x - 1}\right)}{2 x - 1}\right) = \infty$$
- 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
$$\left(-\infty, - \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2} - \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2}\right] \cup \left[- \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2} + \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2}, \infty\right)$$
Convex at the intervals
$$\left[- \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2} - \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2}, - \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2} + \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(- \frac{2 x \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} - 9} + 6 x - 1}{x^{2} - 9} + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{2 x - 1}\right)}{2 x - 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2} + \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2}$$
$$x_{2} = - \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2} - \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -3$$
$$x_{2} = 3$$
$$\lim_{x \to -3^-}\left(\frac{2 \left(- \frac{2 x \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} - 9} + 6 x - 1}{x^{2} - 9} + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{2 x - 1}\right)}{2 x - 1}\right) = \infty$$
$$\lim_{x \to -3^+}\left(\frac{2 \left(- \frac{2 x \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} - 9} + 6 x - 1}{x^{2} - 9} + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{2 x - 1}\right)}{2 x - 1}\right) = \infty$$
- limits are equal, then skip the corresponding point
$$\lim_{x \to 3^-}\left(\frac{2 \left(- \frac{2 x \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} - 9} + 6 x - 1}{x^{2} - 9} + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{2 x - 1}\right)}{2 x - 1}\right) = \infty$$
$$\lim_{x \to 3^+}\left(\frac{2 \left(- \frac{2 x \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{x^{2} - 9} - \frac{- \frac{4 x^{2} \left(2 x - 1\right)}{x^{2} - 9} + 6 x - 1}{x^{2} - 9} + \frac{2 \left(\frac{x \left(2 x - 1\right)}{x^{2} - 9} - 1\right)}{2 x - 1}\right)}{2 x - 1}\right) = \infty$$
- 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
$$\left(-\infty, - \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2} - \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2}\right] \cup \left[- \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2} + \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2}, \infty\right)$$
Convex at the intervals
$$\left[- \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2} - \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2}, - \frac{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}{2} + \frac{1}{2} + \frac{\sqrt{- \frac{140}{3} - 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}} - \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + \frac{35}{\sqrt{- \frac{70}{3} + \frac{1225}{72 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}} + 2 \sqrt[3]{\frac{1225 \sqrt{2706}}{96} + \frac{1147825}{1728}}}}}}{2}\right]$$
Vertical asymptotes
Have:
$$x_{1} = -3$$
$$x_{2} = 3$$
$$x_{1} = -3$$
$$x_{2} = 3$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \log{\left(\frac{2 x - 1}{x^{2} - 9} \right)} = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \log{\left(\frac{2 x - 1}{x^{2} - 9} \right)} = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \log{\left(\frac{2 x - 1}{x^{2} - 9} \right)} = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \log{\left(\frac{2 x - 1}{x^{2} - 9} \right)} = -\infty$$
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 log((2*x - 1)/(x^2 - 9)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{2 x - 1}{x^{2} - 9} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{2 x - 1}{x^{2} - 9} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{2 x - 1}{x^{2} - 9} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{2 x - 1}{x^{2} - 9} \right)}}{x}\right) = 0$$
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:
$$\log{\left(\frac{2 x - 1}{x^{2} - 9} \right)} = \log{\left(\frac{- 2 x - 1}{x^{2} - 9} \right)}$$
- No
$$\log{\left(\frac{2 x - 1}{x^{2} - 9} \right)} = - \log{\left(\frac{- 2 x - 1}{x^{2} - 9} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\log{\left(\frac{2 x - 1}{x^{2} - 9} \right)} = \log{\left(\frac{- 2 x - 1}{x^{2} - 9} \right)}$$
- No
$$\log{\left(\frac{2 x - 1}{x^{2} - 9} \right)} = - \log{\left(\frac{- 2 x - 1}{x^{2} - 9} \right)}$$
- No
so, the function
not is
neither even, nor odd