summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y =…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=ln (x + 5)-4 )\na. the ( y )-intercept of ( f ) is ( y=ln (5)-4 ).\n(type an exact answer. use a comma to separate answers as needed.)\nb. the function ( f ) has no ( y )-intercept.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is increasing on the subinterval(s) ( (-5, infty) ).\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function ( f ) is never increasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is decreasing on the subinterval(s) \n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function ( f ) is never decreasing.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=ln (x + 5)-4 )\na. the ( y )-intercept of ( f ) is ( y=ln (5)-4 ).\n(type an exact answer. use a comma to separate answers as needed.)\nb. the function ( f ) has no ( y )-intercept.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is increasing on the subinterval(s) ( (-5, infty) ).\n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function ( f ) is never increasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) is decreasing on the subinterval(s) \n(type your answer in interval notation. use a comma to separate answers as needed.)\nb. the function ( f ) is never decreasing.

Answer

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.