Mister Exam
Graphing y = ln*1/(x^2-4x+8)
The solution
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log(1)
f(x) = ------------
2
x - 4*x + 8
$$f{\left(x \right)} = \frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8}$$
f = log(1)/(x^2 - 4*x + 8)
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:
$$\frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8} = 0$$
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:
$$\frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8} = 0$$
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
$$\frac{\left(4 - 2 x\right) \log{\left(1 \right)}}{\left(\left(x^{2} - 4 x\right) + 8\right)^{2}} = 0$$
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
$$\frac{\left(4 - 2 x\right) \log{\left(1 \right)}}{\left(\left(x^{2} - 4 x\right) + 8\right)^{2}} = 0$$
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
$$\frac{2 \left(\frac{4 \left(x - 2\right)^{2}}{x^{2} - 4 x + 8} - 1\right) \log{\left(1 \right)}}{\left(x^{2} - 4 x + 8\right)^{2}} = 0$$
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
$$\frac{2 \left(\frac{4 \left(x - 2\right)^{2}}{x^{2} - 4 x + 8} - 1\right) \log{\left(1 \right)}}{\left(x^{2} - 4 x + 8\right)^{2}} = 0$$
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
$$\lim_{x \to -\infty}\left(\frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(1)/(x^2 - 4*x + 8), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(1 \right)}}{x \left(\left(x^{2} - 4 x\right) + 8\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(1 \right)}}{x \left(\left(x^{2} - 4 x\right) + 8\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(1 \right)}}{x \left(\left(x^{2} - 4 x\right) + 8\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(1 \right)}}{x \left(\left(x^{2} - 4 x\right) + 8\right)}\right) = 0$$
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:
$$\frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8} = \frac{\log{\left(1 \right)}}{x^{2} + 4 x + 8}$$
- No
$$\frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8} = - \frac{\log{\left(1 \right)}}{x^{2} + 4 x + 8}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8} = \frac{\log{\left(1 \right)}}{x^{2} + 4 x + 8}$$
- No
$$\frac{\log{\left(1 \right)}}{\left(x^{2} - 4 x\right) + 8} = - \frac{\log{\left(1 \right)}}{x^{2} + 4 x + 8}$$
- No
so, the function
not is
neither even, nor odd