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 ( 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) ( (-5, infty) ).
(type your answer in interval notation. use a comma to separate answers as needed.)
b. the function ( f ) is never increasing.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function ( f ) is decreasing on the subinterval(s)
(type your answer in interval notation. use a comma to separate answers as needed.)
b. the function ( f ) is never decreasing.

Explanation:

Step1: Find the y - intercept

The y - intercept occurs when \(x = 0\). Substitute \(x=0\) into \(y=\ln(x + 5)-4\), we get \(y=\ln(0 + 5)-4=\ln(5)-4\).

Step2: Determine the increasing/decreasing nature

The derivative of \(y = f(x)=\ln(x + 5)-4\) is \(y^\prime=\frac{1}{x + 5}\).
For the function to be increasing, \(y^\prime>0\). Since \(\frac{1}{x + 5}>0\) when \(x+5>0\) (because the denominator \(x + 5\) and the numerator \(1\) have the same sign for \(y^\prime>0\)), i.e., \(x>-5\).
For the function to be decreasing, \(y^\prime<0\). But \(\frac{1}{x + 5}<0\) has no solution because the numerator \(1>0\) and the denominator \(x + 5>0\) for the domain \(x>-5\) (the domain of \(y = \ln(x + 5)-4\) is \(x>-5\) since the argument of the logarithm \(x + 5>0\)).

Answer:

For the y - intercept: A. The y - intercept of \(f\) is \(y=\ln(5)-4\).
For the increasing interval: A. The function \(f\) is increasing on the subinterval(s) \((-5,\infty)\).
For the decreasing interval: B. The function \(f\) is never decreasing.