Mister Exam
Graphing y = log(x-4,1/3)/3
The solution
You have entered
[src]
/ 41 \
log|x - ----|
\ 10*3/
f(x) = -------------
3
$$f{\left(x \right)} = \frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3}$$
f = log(x - 41/30)/3
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:
$$\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{71}{30}$$
Numerical solution
$$x_{1} = 2.36666666666667$$
so we need to solve the equation:
$$\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{71}{30}$$
Numerical solution
$$x_{1} = 2.36666666666667$$
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
$$\frac{1}{3 \left(x - \frac{41}{3 \cdot 10}\right)} = 0$$
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
$$\frac{1}{3 \left(x - \frac{41}{3 \cdot 10}\right)} = 0$$
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
$$- \frac{1}{3 \left(x - \frac{41}{30}\right)^{2}} = 0$$
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
$$- \frac{1}{3 \left(x - \frac{41}{30}\right)^{2}} = 0$$
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
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3}\right) = \infty$$
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 log(x - 41/30)/3, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3 x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3 x}\right) = 0$$
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:
$$\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3} = \frac{\log{\left(- x - \frac{41}{30} \right)}}{3}$$
- No
$$\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3} = - \frac{\log{\left(- x - \frac{41}{30} \right)}}{3}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3} = \frac{\log{\left(- x - \frac{41}{30} \right)}}{3}$$
- No
$$\frac{\log{\left(x - \frac{41}{3 \cdot 10} \right)}}{3} = - \frac{\log{\left(- x - \frac{41}{30} \right)}}{3}$$
- No
so, the function
not is
neither even, nor odd