Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of y = f(x).

f(x) = ln(x + 5) - 4

a. the x - intercept(s) of f is x = e^{4}-5
(type an exact answer. use a comma to separate answers as needed.)
b. the function f has no x - intercepts.

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the y - intercept of f is y = ln(5)-4
(type an exact answer. use a comma to separate answers as needed.)
b. the function f has no y - intercept.

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the function f is increasing on the subinterval(s)
(type your answer in interval notation. use a comma to separate answers as needed.)
b. the function f is never increasing.

Explanation:

Step1: Find the domain

For \(y = \ln(x + 5)-4\), the argument of the logarithm must be positive. So \(x+5>0\), which gives \(x>- 5\). The domain is \((-5,\infty)\).

Step2: Analyze the derivative

The derivative of \(y=f(x)=\ln(x + 5)-4\) is \(y^\prime=\frac{1}{x + 5}\).
Since \(x>-5\), then \(y^\prime=\frac{1}{x + 5}>0\) for all \(x\in(-5,\infty)\).

Answer:

A. The function \(f\) is increasing on the sub - interval \((-5,\infty)\)