Mister Exam
Graphing y = log((1-1)/x^2)
The solution
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f(x) = log|--|
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$$f{\left(x \right)} = \log{\left(\frac{0}{x^{2}} \right)}$$
f = log(0/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:
$$\log{\left(\frac{0}{x^{2}} \right)} = 0$$
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:
$$\log{\left(\frac{0}{x^{2}} \right)} = 0$$
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
$$0 = 0$$
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
$$0 = 0$$
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
$$0 = 0$$
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
$$0 = 0$$
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
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
False
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
False
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(0/x^2), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
False
Let's take the limit
so,
inclined asymptote on the left doesn’t exist
False
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
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:
$$\log{\left(\frac{0}{x^{2}} \right)} = \log{\left(\frac{0}{x^{2}} \right)}$$
- Yes
$$\log{\left(\frac{0}{x^{2}} \right)} = - \log{\left(\frac{0}{x^{2}} \right)}$$
- No
so, the function
is
even
So, check:
$$\log{\left(\frac{0}{x^{2}} \right)} = \log{\left(\frac{0}{x^{2}} \right)}$$
- Yes
$$\log{\left(\frac{0}{x^{2}} \right)} = - \log{\left(\frac{0}{x^{2}} \right)}$$
- No
so, the function
is
even