Mister Exam
Graphing y = |log10(x^2)-1|
The solution
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| / 2\ |
|log\x / |
f(x) = |------- - 1|
|log(10) |
$$f{\left(x \right)} = \left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right|$$
f = Abs(log(x^2)/log(10) - 1)
The graph of the function
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:
$$\left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 3.16227766016838$$
$$x_{2} = -3.16227766016838$$
so we need to solve the equation:
$$\left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 3.16227766016838$$
$$x_{2} = -3.16227766016838$$
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
$$\frac{2 \left(\frac{d}{d x} \operatorname{sign}{\left(\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1 \right)} - \frac{\operatorname{sign}{\left(\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1 \right)}}{x}\right)}{x \log{\left(10 \right)}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(\frac{d}{d x} \operatorname{sign}{\left(\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1 \right)} - \frac{\operatorname{sign}{\left(\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1 \right)}}{x}\right)}{x \log{\left(10 \right)}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty} \left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right| = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty} \left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right| = \infty$$
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 Abs(log(x^2)/log(10) - 1), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right|}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right|}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right|}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right|}{x}\right)$$
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:
$$\left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right| = \left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right|$$
- Yes
$$\left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right| = - \left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right|$$
- No
so, the function
is
even
So, check:
$$\left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right| = \left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right|$$
- Yes
$$\left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right| = - \left|{\frac{\log{\left(x^{2} \right)}}{\log{\left(10 \right)}} - 1}\right|$$
- No
so, the function
is
even