Mister Exam
Graphing y = 2ln((x+)3/x)-3
The solution
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/x*3\
f(x) = 2*log|---| - 3
\ x /
$$f{\left(x \right)} = 2 \log{\left(\frac{3 x}{x} \right)} - 3$$
f = 2*log((3*x)/x) - 3
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:
$$2 \log{\left(\frac{3 x}{x} \right)} - 3 = 0$$
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:
$$2 \log{\left(\frac{3 x}{x} \right)} - 3 = 0$$
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
$$\lim_{x \to -\infty}\left(2 \log{\left(\frac{3 x}{x} \right)} - 3\right) = -3 + 2 \log{\left(3 \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -3 + 2 \log{\left(3 \right)}$$
$$\lim_{x \to \infty}\left(2 \log{\left(\frac{3 x}{x} \right)} - 3\right) = -3 + 2 \log{\left(3 \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -3 + 2 \log{\left(3 \right)}$$
$$\lim_{x \to -\infty}\left(2 \log{\left(\frac{3 x}{x} \right)} - 3\right) = -3 + 2 \log{\left(3 \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = -3 + 2 \log{\left(3 \right)}$$
$$\lim_{x \to \infty}\left(2 \log{\left(\frac{3 x}{x} \right)} - 3\right) = -3 + 2 \log{\left(3 \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = -3 + 2 \log{\left(3 \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*log((x*3)/x) - 3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{2 \log{\left(\frac{3 x}{x} \right)} - 3}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 \log{\left(\frac{3 x}{x} \right)} - 3}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{2 \log{\left(\frac{3 x}{x} \right)} - 3}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{2 \log{\left(\frac{3 x}{x} \right)} - 3}{x}\right) = 0$$
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:
$$2 \log{\left(\frac{3 x}{x} \right)} - 3 = -3 + 2 \log{\left(3 \right)}$$
- No
$$2 \log{\left(\frac{3 x}{x} \right)} - 3 = 3 - 2 \log{\left(3 \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$2 \log{\left(\frac{3 x}{x} \right)} - 3 = -3 + 2 \log{\left(3 \right)}$$
- No
$$2 \log{\left(\frac{3 x}{x} \right)} - 3 = 3 - 2 \log{\left(3 \right)}$$
- No
so, the function
not is
neither even, nor odd