Answer:
d. the domain is (1, infinity) and the range is all real numbers
Step-by-step explanation:
The domain of a logarithmic function
[tex]f(x)=\ln(x)[/tex]
is the set of all positive numbers
[tex]D:\ x>0\Rightarrow x\in\mathbb{R}^+[/tex]
The range of a logarithmic function
[tex]f(x)=\ln(x)[/tex]
is the set of all real numbers
[tex]R:\ y\in\mathbb{R}[/tex]
We have:
[tex]y=\ln(x-1)+2[/tex]
DOMAIN
[tex]x-1>0[/tex] add 1 to both sides
[tex]x-1+1>0+1\\\\x>1[/tex]
[tex]D:x>0\Rightarrow x\in(1,\ \infty)[/tex]
RANGE
[tex]f(x)=\ln(x)\to f(x)+2=\ln(x)+2[/tex]
The graph shifted 2 units up. The range no change.
[tex]R:\ y\in\mathbb{R}[/tex]
