Question

1. choose the best descriptor for the graph of f(x) = ln(x - 5). (2 points)

a. it increases rapidly, goes through the point (5, 1), then increases gradually.
b. it increases slowly, goes through the point (5, 1), then increases rapidly.
c. it increases slowly, goes through the point (6, 0), then increases gradually.
d. it increases rapidly, goes through the point (6, 0), then increases gradually.

Explanation:

Step1: Recall domain of natural - log function

The domain of $y = \ln(u)$ is $u>0$. For $y=\ln(x - 5)$, we have $x-5>0$, so $x>5$.

Step2: Find the x - intercept

Set $y = 0$. Then $\ln(x - 5)=0$. Since $\ln1 = 0$, we have $x-5 = 1$, so $x=6$. The graph passes through the point $(6,0)$.

Step3: Analyze the behavior of the natural - log function

The function $y=\ln(x)$ has a slow - increasing behavior. The function $y=\ln(x - 5)$ is a horizontal shift of $y=\ln(x)$ to the right by 5 units and also has a slow - increasing behavior.

Answer:

c. It increases slowly, goes through the point $(6,0)$, then increases gradually.