Mister Exam
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-
How to use it?
- Graphing y =:
-
(x^2+3)/(x-1)
- 3x-x^2
- x*sin(1/x)
- xsin1/x
-
Identical expressions
- -3log2x+ one
- minus 3 logarithm of 2x plus 1
- minus 3 logarithm of 2x plus one
-
Similar expressions
- -3log2x-1
- 3log2x+1
Graphing y = -3log2x+1
The solution
You have entered
[src]
f(x) = -3*log(2*x) + 1
$$f{\left(x \right)} = 1 - 3 \log{\left(2 x \right)}$$
f = 1 - 3*log(2*x)
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:
$$1 - 3 \log{\left(2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{e^{\frac{1}{3}}}{2}$$
Numerical solution
$$x_{1} = 0.697806212543045$$
so we need to solve the equation:
$$1 - 3 \log{\left(2 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{e^{\frac{1}{3}}}{2}$$
Numerical solution
$$x_{1} = 0.697806212543045$$
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
$$- \frac{3}{x} = 0$$
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
$$- \frac{3}{x} = 0$$
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
$$\frac{3}{x^{2}} = 0$$
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
$$\frac{3}{x^{2}} = 0$$
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
$$\lim_{x \to -\infty}\left(1 - 3 \log{\left(2 x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(1 - 3 \log{\left(2 x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(1 - 3 \log{\left(2 x \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(1 - 3 \log{\left(2 x \right)}\right) = -\infty$$
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(2*x) + 1, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{1 - 3 \log{\left(2 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{1 - 3 \log{\left(2 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{1 - 3 \log{\left(2 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{1 - 3 \log{\left(2 x \right)}}{x}\right) = 0$$
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:
$$1 - 3 \log{\left(2 x \right)} = 1 - 3 \log{\left(- 2 x \right)}$$
- No
$$1 - 3 \log{\left(2 x \right)} = 3 \log{\left(- 2 x \right)} - 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$1 - 3 \log{\left(2 x \right)} = 1 - 3 \log{\left(- 2 x \right)}$$
- No
$$1 - 3 \log{\left(2 x \right)} = 3 \log{\left(- 2 x \right)} - 1$$
- No
so, the function
not is
neither even, nor odd