Graphing y = (x+ln(x))/(x-4)

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
$$\frac{- \frac{2 \left(1 + \frac{1}{x}\right)}{x - 4} + \frac{2 \left(x + \log{\left(x \right)}\right)}{\left(x - 4\right)^{2}} - \frac{1}{x^{2}}}{x - 4} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 38783.8985529685$$
$$x_{2} = 45808.020325951$$
$$x_{3} = 29719.8418978506$$
$$x_{4} = 37778.7731403642$$
$$x_{5} = 31737.8127530236$$
$$x_{6} = 40792.827428034$$
$$x_{7} = 30729.1207207176$$
$$x_{8} = 0.8114375561742$$
$$x_{9} = 47811.4762989485$$
$$x_{10} = 46809.925612916$$
$$x_{11} = 34760.5793720106$$
$$x_{12} = 28709.9529706705$$
$$x_{13} = 35767.1290742184$$
$$x_{14} = 48812.6813980915$$
$$x_{15} = 44805.7509925326$$
$$x_{16} = 42800.0800645983$$
$$x_{17} = 43803.1077057361$$
$$x_{18} = 41796.6571384727$$
$$x_{19} = 32745.9397987124$$
$$x_{20} = 36773.1883391132$$
$$x_{21} = 27699.4289684316$$
$$x_{22} = 33753.5222833995$$
$$x_{23} = 39788.5788225481$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 4$$

$$\lim_{x \to 4^-}\left(\frac{- \frac{2 \left(1 + \frac{1}{x}\right)}{x - 4} + \frac{2 \left(x + \log{\left(x \right)}\right)}{\left(x - 4\right)^{2}} - \frac{1}{x^{2}}}{x - 4}\right) = -\infty$$
$$\lim_{x \to 4^+}\left(\frac{- \frac{2 \left(1 + \frac{1}{x}\right)}{x - 4} + \frac{2 \left(x + \log{\left(x \right)}\right)}{\left(x - 4\right)^{2}} - \frac{1}{x^{2}}}{x - 4}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = 4$$
- 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
$$\left(-\infty, 0.8114375561742\right]$$
Convex at the intervals
$$\left[0.8114375561742, \infty\right)$$