Mister Exam
Graphing y = log10(x-1/x+2)
The solution
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/ 1 \
log|x - - + 2|
\ x /
f(x) = --------------
log(10)
$$f{\left(x \right)} = \frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}}$$
f = log(x - 1/x + 2)/log(10)
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:
$$\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$x_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
Numerical solution
$$x_{1} = 0.618033988749895$$
$$x_{2} = -1.61803398874989$$
so we need to solve the equation:
$$\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$x_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
Numerical solution
$$x_{1} = 0.618033988749895$$
$$x_{2} = -1.61803398874989$$
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
$$\frac{1 + \frac{1}{x^{2}}}{\left(\left(x - \frac{1}{x}\right) + 2\right) \log{\left(10 \right)}} = 0$$
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
$$\frac{1 + \frac{1}{x^{2}}}{\left(\left(x - \frac{1}{x}\right) + 2\right) \log{\left(10 \right)}} = 0$$
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
$$- \frac{\frac{\left(1 + \frac{1}{x^{2}}\right)^{2}}{x + 2 - \frac{1}{x}} + \frac{2}{x^{3}}}{\left(x + 2 - \frac{1}{x}\right) \log{\left(10 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -1$$
$$x_{2} = - \frac{\sqrt[3]{8 + 6 \sqrt{78}}}{3} + \frac{1}{3} + \frac{14}{3 \sqrt[3]{8 + 6 \sqrt{78}}}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{\frac{\left(1 + \frac{1}{x^{2}}\right)^{2}}{x + 2 - \frac{1}{x}} + \frac{2}{x^{3}}}{\left(x + 2 - \frac{1}{x}\right) \log{\left(10 \right)}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(- \frac{\frac{\left(1 + \frac{1}{x^{2}}\right)^{2}}{x + 2 - \frac{1}{x}} + \frac{2}{x^{3}}}{\left(x + 2 - \frac{1}{x}\right) \log{\left(10 \right)}}\right) = \infty$$
- 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
$$\left[-1, - \frac{\sqrt[3]{8 + 6 \sqrt{78}}}{3} + \frac{1}{3} + \frac{14}{3 \sqrt[3]{8 + 6 \sqrt{78}}}\right]$$
Convex at the intervals
$$\left(-\infty, -1\right] \cup \left[- \frac{\sqrt[3]{8 + 6 \sqrt{78}}}{3} + \frac{1}{3} + \frac{14}{3 \sqrt[3]{8 + 6 \sqrt{78}}}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\frac{\left(1 + \frac{1}{x^{2}}\right)^{2}}{x + 2 - \frac{1}{x}} + \frac{2}{x^{3}}}{\left(x + 2 - \frac{1}{x}\right) \log{\left(10 \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -1$$
$$x_{2} = - \frac{\sqrt[3]{8 + 6 \sqrt{78}}}{3} + \frac{1}{3} + \frac{14}{3 \sqrt[3]{8 + 6 \sqrt{78}}}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{\frac{\left(1 + \frac{1}{x^{2}}\right)^{2}}{x + 2 - \frac{1}{x}} + \frac{2}{x^{3}}}{\left(x + 2 - \frac{1}{x}\right) \log{\left(10 \right)}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(- \frac{\frac{\left(1 + \frac{1}{x^{2}}\right)^{2}}{x + 2 - \frac{1}{x}} + \frac{2}{x^{3}}}{\left(x + 2 - \frac{1}{x}\right) \log{\left(10 \right)}}\right) = \infty$$
- 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
$$\left[-1, - \frac{\sqrt[3]{8 + 6 \sqrt{78}}}{3} + \frac{1}{3} + \frac{14}{3 \sqrt[3]{8 + 6 \sqrt{78}}}\right]$$
Convex at the intervals
$$\left(-\infty, -1\right] \cup \left[- \frac{\sqrt[3]{8 + 6 \sqrt{78}}}{3} + \frac{1}{3} + \frac{14}{3 \sqrt[3]{8 + 6 \sqrt{78}}}, \infty\right)$$
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
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}}\right) = \infty$$
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 log(x - 1/x + 2)/log(10), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{x \log{\left(10 \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{x \log{\left(10 \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{x \log{\left(10 \right)}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{x \log{\left(10 \right)}}\right) = 0$$
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:
$$\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}} = \frac{\log{\left(- x + 2 + \frac{1}{x} \right)}}{\log{\left(10 \right)}}$$
- No
$$\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}} = - \frac{\log{\left(- x + 2 + \frac{1}{x} \right)}}{\log{\left(10 \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}} = \frac{\log{\left(- x + 2 + \frac{1}{x} \right)}}{\log{\left(10 \right)}}$$
- No
$$\frac{\log{\left(\left(x - \frac{1}{x}\right) + 2 \right)}}{\log{\left(10 \right)}} = - \frac{\log{\left(- x + 2 + \frac{1}{x} \right)}}{\log{\left(10 \right)}}$$
- No
so, the function
not is
neither even, nor odd