Question

question 5
2 pts
for the function ( y = ln(x - 1)+2 ), which of the following statements is true?
a. the domain is ( (1,infty) ), and the range is ( 2,infty) ).
b. the domain is ( (1,infty) ), and the range is all real numbers.
c. the domain is ( (-1,infty) ), and the range is all real numbers.
d. the domain is all real numbers, and the range is ( 2,infty) ).

Explanation:

Step1: Find the domain

For the function \(y = \ln(x - 1)+2\), the argument of the natural logarithm \(x-1>0\) (since the domain of \(y = \ln(u)\) is \(u>0\)).
Solving \(x - 1>0\) gives \(x>1\). So the domain is \((1,\infty)\).

Step2: Find the range

The range of the basic natural - logarithm function \(y=\ln(u)\) is \((-\infty,\infty)\).
For the function \(y=\ln(x - 1)+2\), it is a vertical shift of \(y = \ln(x-1)\) by 2 units. Since shifting a function vertically does not restrict the range (only shifts it), the range of \(y=\ln(x - 1)+2\) is still \((-\infty,\infty)\) (all real numbers).

Answer:

B. The domain is \((1,\infty)\), and the range is all real numbers.