Mister Exam
Graphing y = ln*(x+4)/(x-2)
The solution
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log(x + 4)
f(x) = ----------
x - 2
$$f{\left(x \right)} = \frac{\log{\left(x + 4 \right)}}{x - 2}$$
f = log(x + 4)/(x - 2)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 2$$
$$x_{1} = 2$$
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:
$$\frac{\log{\left(x + 4 \right)}}{x - 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3$$
Numerical solution
$$x_{1} = -3$$
so we need to solve the equation:
$$\frac{\log{\left(x + 4 \right)}}{x - 2} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = -3$$
Numerical solution
$$x_{1} = -3$$
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
$$\frac{1}{\left(x - 2\right) \left(x + 4\right)} - \frac{\log{\left(x + 4 \right)}}{\left(x - 2\right)^{2}} = 0$$
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
$$\frac{1}{\left(x - 2\right) \left(x + 4\right)} - \frac{\log{\left(x + 4 \right)}}{\left(x - 2\right)^{2}} = 0$$
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
$$\frac{- \frac{1}{\left(x + 4\right)^{2}} - \frac{2}{\left(x - 2\right) \left(x + 4\right)} + \frac{2 \log{\left(x + 4 \right)}}{\left(x - 2\right)^{2}}}{x - 2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 52579.58620952$$
$$x_{2} = 40259.2239485916$$
$$x_{3} = 27732.3756214626$$
$$x_{4} = 31175.4996932754$$
$$x_{5} = 57023.1752429365$$
$$x_{6} = 25423.0740467586$$
$$x_{7} = 35733.2068252472$$
$$x_{8} = 55913.8430109058$$
$$x_{9} = 24263.7470545983$$
$$x_{10} = 46999.5741529783$$
$$x_{11} = 45879.7815488777$$
$$x_{12} = 30030.3664344542$$
$$x_{13} = 34596.9830552026$$
$$x_{14} = 42511.9858242385$$
$$x_{15} = 32318.2354224996$$
$$x_{16} = 48118.0458630616$$
$$x_{17} = -2.26148779748038$$
$$x_{18} = 39130.3889084217$$
$$x_{19} = 44758.620320137$$
$$x_{20} = 26579.2215804665$$
$$x_{21} = 43636.0397404855$$
$$x_{22} = 36867.4585864673$$
$$x_{23} = 53692.0799725151$$
$$x_{24} = 28882.7056429339$$
$$x_{25} = 51465.9728320884$$
$$x_{26} = 50351.2039481891$$
$$x_{27} = 41386.4010329619$$
$$x_{28} = 33458.6931594049$$
$$x_{29} = 49235.2416517488$$
$$x_{30} = 37999.8255966667$$
$$x_{31} = 54803.4881441841$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 2$$
$$\lim_{x \to 2^-}\left(\frac{- \frac{1}{\left(x + 4\right)^{2}} - \frac{2}{\left(x - 2\right) \left(x + 4\right)} + \frac{2 \log{\left(x + 4 \right)}}{\left(x - 2\right)^{2}}}{x - 2}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{- \frac{1}{\left(x + 4\right)^{2}} - \frac{2}{\left(x - 2\right) \left(x + 4\right)} + \frac{2 \log{\left(x + 4 \right)}}{\left(x - 2\right)^{2}}}{x - 2}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 2$$
- is an inflection point
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left(-\infty, -2.26148779748038\right]$$
Convex at the intervals
$$\left[-2.26148779748038, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \frac{1}{\left(x + 4\right)^{2}} - \frac{2}{\left(x - 2\right) \left(x + 4\right)} + \frac{2 \log{\left(x + 4 \right)}}{\left(x - 2\right)^{2}}}{x - 2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 52579.58620952$$
$$x_{2} = 40259.2239485916$$
$$x_{3} = 27732.3756214626$$
$$x_{4} = 31175.4996932754$$
$$x_{5} = 57023.1752429365$$
$$x_{6} = 25423.0740467586$$
$$x_{7} = 35733.2068252472$$
$$x_{8} = 55913.8430109058$$
$$x_{9} = 24263.7470545983$$
$$x_{10} = 46999.5741529783$$
$$x_{11} = 45879.7815488777$$
$$x_{12} = 30030.3664344542$$
$$x_{13} = 34596.9830552026$$
$$x_{14} = 42511.9858242385$$
$$x_{15} = 32318.2354224996$$
$$x_{16} = 48118.0458630616$$
$$x_{17} = -2.26148779748038$$
$$x_{18} = 39130.3889084217$$
$$x_{19} = 44758.620320137$$
$$x_{20} = 26579.2215804665$$
$$x_{21} = 43636.0397404855$$
$$x_{22} = 36867.4585864673$$
$$x_{23} = 53692.0799725151$$
$$x_{24} = 28882.7056429339$$
$$x_{25} = 51465.9728320884$$
$$x_{26} = 50351.2039481891$$
$$x_{27} = 41386.4010329619$$
$$x_{28} = 33458.6931594049$$
$$x_{29} = 49235.2416517488$$
$$x_{30} = 37999.8255966667$$
$$x_{31} = 54803.4881441841$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 2$$
$$\lim_{x \to 2^-}\left(\frac{- \frac{1}{\left(x + 4\right)^{2}} - \frac{2}{\left(x - 2\right) \left(x + 4\right)} + \frac{2 \log{\left(x + 4 \right)}}{\left(x - 2\right)^{2}}}{x - 2}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{- \frac{1}{\left(x + 4\right)^{2}} - \frac{2}{\left(x - 2\right) \left(x + 4\right)} + \frac{2 \log{\left(x + 4 \right)}}{\left(x - 2\right)^{2}}}{x - 2}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 2$$
- is an inflection point
Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left(-\infty, -2.26148779748038\right]$$
Convex at the intervals
$$\left[-2.26148779748038, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = 2$$
$$x_{1} = 2$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x + 4 \right)}}{x - 2}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x + 4 \right)}}{x - 2}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x + 4 \right)}}{x - 2}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x + 4 \right)}}{x - 2}\right) = 0$$
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(x + 4)/(x - 2), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x + 4 \right)}}{x \left(x - 2\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x + 4 \right)}}{x \left(x - 2\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x + 4 \right)}}{x \left(x - 2\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x + 4 \right)}}{x \left(x - 2\right)}\right) = 0$$
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:
$$\frac{\log{\left(x + 4 \right)}}{x - 2} = \frac{\log{\left(4 - x \right)}}{- x - 2}$$
- No
$$\frac{\log{\left(x + 4 \right)}}{x - 2} = - \frac{\log{\left(4 - x \right)}}{- x - 2}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\log{\left(x + 4 \right)}}{x - 2} = \frac{\log{\left(4 - x \right)}}{- x - 2}$$
- No
$$\frac{\log{\left(x + 4 \right)}}{x - 2} = - \frac{\log{\left(4 - x \right)}}{- x - 2}$$
- No
so, the function
not is
neither even, nor odd