To determine the domain and the vertical asymptote of the function [tex]\( g(x) = \ln(7 - x) \)[/tex], let's examine the properties of the natural logarithm function.
### Domain:
The natural logarithm function, [tex]\( \ln(y) \)[/tex], is defined only when its argument [tex]\( y \)[/tex] is positive. Therefore, for [tex]\( g(x) = \ln(7 - x) \)[/tex] to be defined, the argument [tex]\( 7 - x \)[/tex] must be greater than 0.
[tex]\[ 7 - x > 0 \][/tex]
Solving this inequality for [tex]\( x \)[/tex]:
[tex]\[ x < 7 \][/tex]
Hence, the domain of the function [tex]\( g(x) = \ln(7 - x) \)[/tex] is all real numbers [tex]\( x \)[/tex] that are less than 7. In interval notation, this is:
[tex]\[ \boxed{(-\infty, 7)} \][/tex]
### Vertical Asymptote:
A vertical asymptote occurs where the value inside the natural logarithm approaches zero, since the natural logarithm of zero is undefined and as it approaches zero, [tex]\( \ln(y) \)[/tex] tends to negative infinity.
Set the argument of the natural logarithm to zero:
[tex]\[ 7 - x = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = 7 \][/tex]
Thus, the function [tex]\( g(x) = \ln(7 - x) \)[/tex] has a vertical asymptote at:
[tex]\[ \boxed{x = 7} \][/tex]
In summary:
- The domain of [tex]\( g(x) = \ln(7 - x) \)[/tex] is [tex]\( \boxed{(-\infty, 7)} \)[/tex].
- The vertical asymptote of the function is at [tex]\( \boxed{x = 7} \)[/tex].
