the movement of the progress bar may be uneven because questions can be worth more or less (including zero)…

Explanation:

Step1: Find the domain

The argument of the natural - logarithm function must be positive. So (x + 1>0), which gives (x>- 1). The domain is ((-1,\infty)).

Step2: Find the y - intercept

Set (x = 0). Then (f(0)=2\ln(0 + 1)=2\ln(1)=0). So the y - intercept is at the point ((0,0)).

Step3: Analyze the end - behavior

As (x\to - 1^{+}), (y = 2\ln(x + 1)\to-\infty). As (x\to\infty), (y = 2\ln(x + 1)\to\infty). The function is an increasing function since the derivative (f^\prime(x)=\frac{2}{x + 1}>0) for (x>-1).

Answer:

The graph has a vertical asymptote at (x=-1), passes through the point ((0,0)), and is an increasing function for (x > - 1).