Mister Exam
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-
How to use it?
- Graphing y =:
- (x+2)^2/(x+1)
- x^3-6x^2+9x-5
- x^3+6x^2+9x-3
- x^3+6x^2-4
- Derivative of:
-
log(1+1/(x^2))
-
Identical expressions
- log(one + one /(x^ two))
- logarithm of (1 plus 1 divide by (x squared ))
- logarithm of (one plus one divide by (x to the power of two))
- log(1+1/(x2))
- log1+1/x2
- log(1+1/(x²))
- log(1+1/(x to the power of 2))
- log1+1/x^2
- log(1+1 divide by (x^2))
-
Similar expressions
- log(1-1/(x^2))
Graphing y = log(1+1/(x^2))
The solution
You have entered
[src]
/ 1 \
f(x) = log|1 + --|
| 2|
\ x /
$$f{\left(x \right)} = \log{\left(1 + \frac{1}{x^{2}} \right)}$$
f = log(1 + 1/(x^2))
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:
$$\log{\left(1 + \frac{1}{x^{2}} \right)} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
so we need to solve the equation:
$$\log{\left(1 + \frac{1}{x^{2}} \right)} = 0$$
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
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
$$- \frac{2}{x^{3} \left(1 + \frac{1}{x^{2}}\right)} = 0$$
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
$$- \frac{2}{x^{3} \left(1 + \frac{1}{x^{2}}\right)} = 0$$
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
$$\frac{2 \left(3 - \frac{2}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{4} \left(1 + \frac{1}{x^{2}}\right)} = 0$$
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
$$\frac{2 \left(3 - \frac{2}{x^{2} \left(1 + \frac{1}{x^{2}}\right)}\right)}{x^{4} \left(1 + \frac{1}{x^{2}}\right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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
$$\lim_{x \to -\infty} \log{\left(1 + \frac{1}{x^{2}} \right)} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty} \log{\left(1 + \frac{1}{x^{2}} \right)} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty} \log{\left(1 + \frac{1}{x^{2}} \right)} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty} \log{\left(1 + \frac{1}{x^{2}} \right)} = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(1 + 1/(x^2)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(1 + \frac{1}{x^{2}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(1 + \frac{1}{x^{2}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(1 + \frac{1}{x^{2}} \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(1 + \frac{1}{x^{2}} \right)}}{x}\right) = 0$$
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:
$$\log{\left(1 + \frac{1}{x^{2}} \right)} = \log{\left(1 + \frac{1}{x^{2}} \right)}$$
- Yes
$$\log{\left(1 + \frac{1}{x^{2}} \right)} = - \log{\left(1 + \frac{1}{x^{2}} \right)}$$
- No
so, the function
is
even
So, check:
$$\log{\left(1 + \frac{1}{x^{2}} \right)} = \log{\left(1 + \frac{1}{x^{2}} \right)}$$
- Yes
$$\log{\left(1 + \frac{1}{x^{2}} \right)} = - \log{\left(1 + \frac{1}{x^{2}} \right)}$$
- No
so, the function
is
even