Mister Exam
Graphing y = 3ln((x/(x-3))-1)
The solution
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/ x \
f(x) = 3*log|----- - 1|
\x - 3 /
$$f{\left(x \right)} = 3 \log{\left(\frac{x}{x - 3} - 1 \right)}$$
f = 3*log(x/(x - 3) - 1)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 3$$
$$x_{1} = 3$$
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:
$$3 \log{\left(\frac{x}{x - 3} - 1 \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 6$$
Numerical solution
$$x_{1} = 6$$
so we need to solve the equation:
$$3 \log{\left(\frac{x}{x - 3} - 1 \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 6$$
Numerical solution
$$x_{1} = 6$$
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
$$\frac{3 \left(- \frac{x}{\left(x - 3\right)^{2}} + \frac{1}{x - 3}\right)}{\frac{x}{x - 3} - 1} = 0$$
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
$$\frac{3 \left(- \frac{x}{\left(x - 3\right)^{2}} + \frac{1}{x - 3}\right)}{\frac{x}{x - 3} - 1} = 0$$
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
$$\frac{3}{\left(x - 3\right)^{2}} = 0$$
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
$$\frac{3}{\left(x - 3\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
$$x_{1} = 3$$
$$x_{1} = 3$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(3 \log{\left(\frac{x}{x - 3} - 1 \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(3 \log{\left(\frac{x}{x - 3} - 1 \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(3 \log{\left(\frac{x}{x - 3} - 1 \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(3 \log{\left(\frac{x}{x - 3} - 1 \right)}\right) = -\infty$$
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 3*log(x/(x - 3) - 1), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{3 \log{\left(\frac{x}{x - 3} - 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{3 \log{\left(\frac{x}{x - 3} - 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{3 \log{\left(\frac{x}{x - 3} - 1 \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{3 \log{\left(\frac{x}{x - 3} - 1 \right)}}{x}\right) = 0$$
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:
$$3 \log{\left(\frac{x}{x - 3} - 1 \right)} = 3 \log{\left(- \frac{x}{- x - 3} - 1 \right)}$$
- No
$$3 \log{\left(\frac{x}{x - 3} - 1 \right)} = - 3 \log{\left(- \frac{x}{- x - 3} - 1 \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$3 \log{\left(\frac{x}{x - 3} - 1 \right)} = 3 \log{\left(- \frac{x}{- x - 3} - 1 \right)}$$
- No
$$3 \log{\left(\frac{x}{x - 3} - 1 \right)} = - 3 \log{\left(- \frac{x}{- x - 3} - 1 \right)}$$
- No
so, the function
not is
neither even, nor odd