To analyze the function [tex]\( g(x) = f(x+4) \)[/tex], we first need to understand the given function [tex]\( f(x) \)[/tex].
The function [tex]\( f(x) = 10^2 \)[/tex] is a constant function, meaning it simplifies to a constant value for all values of [tex]\( x \)[/tex]. Specifically,
[tex]\[ f(x) = 10^2 = 100 \][/tex]
Since [tex]\( f(x) = 100 \)[/tex] for all [tex]\( x \)[/tex], the function [tex]\( g(x) \)[/tex] modifies the input of [tex]\( f \)[/tex] without affecting the output. That is,
[tex]\[ g(x) = f(x + 4) \][/tex]
Substituting into [tex]\( f(x) \)[/tex]:
[tex]\[ g(x) = f(x + 4) = 100 \][/tex]
Thus, [tex]\( g(x) \)[/tex] is also a constant function and is equal to 100 for all [tex]\( x \)[/tex].
Given this, we can assess the given multiple-choice options based on our understanding of [tex]\( g(x) \)[/tex]:
A. Horizontal asymptote of [tex]\( y = 4 \)[/tex]: A horizontal asymptote refers to the value that a function approaches as [tex]\( x \)[/tex] tends toward [tex]\(\pm \infty\)[/tex]. Here, [tex]\( g(x) = 100 \)[/tex], which is a constant function, always equals 100. Therefore, the horizontal asymptote is [tex]\( y = 100 \)[/tex], not [tex]\( y = 4 \)[/tex]. This option is incorrect.
B. [tex]\( x \)[/tex]-intercept at [tex]\( (4, 0) \)[/tex]: The [tex]\( x \)[/tex]-intercept is the value of [tex]\( x \)[/tex] where the function is equal to zero. Since [tex]\( g(x) \)[/tex] is a constant function equal to 100, it never crosses the [tex]\( x \)[/tex]-axis. Therefore, there is no [tex]\( x \)[/tex]-intercept at [tex]\( (4, 0) \)[/tex]. This option is incorrect.
C. Horizontal asymptote of [tex]\( y = 0 \)[/tex]: As explained in option A, the horizontal asymptote of a constant function [tex]\( g(x) = 100 \)[/tex] is [tex]\( y = 100 \)[/tex], not [tex]\( y = 0 \)[/tex]. This option is incorrect.
D. [tex]\( y \)[/tex]-intercept at [tex]\( (0, 4) \)[/tex]: The [tex]\( y \)[/tex]-intercept is the value of the function when [tex]\( x = 0 \)[/tex]. For [tex]\( g(x) = 100 \)[/tex], regardless of the value of [tex]\( x \)[/tex], the function will be 100. Therefore, the [tex]\( y \)[/tex]-intercept is [tex]\( (0, 100) \)[/tex], not [tex]\( (0, 4) \)[/tex]. This option is incorrect.
Given this analysis, none of the provided options (A, B, C, or D) correctly describe a key feature of the function [tex]\( g(x) = f(x + 4) = 100 \)[/tex].
Therefore, the answer is that none of the given options A, B, C, or D, are correct.
