To analyze the function [tex]\( g(x) = f(x + 4) \)[/tex] and determine the key feature associated with it, let's keep in mind that this function represents a horizontal shift of the original function [tex]\( f(x) \)[/tex] by 4 units to the left.
We'll examine each of the multiple-choice options to see if it describes the key feature of the function [tex]\( g(x) \)[/tex]:
A. [tex]$y$[/tex]-intercept at [tex]$(0, 4)$[/tex]:
- The [tex]$y$[/tex]-intercept of a function is the point where [tex]\( x = 0 \)[/tex]. For [tex]\( g(x) = f(x + 4) \)[/tex], the [tex]$y$[/tex]-intercept occurs at [tex]\( g(0) \)[/tex]. This means
[tex]\[
g(0) = f(4).
\][/tex]
Without knowing the specific form of [tex]\( f(x) \)[/tex], we cannot definitively state that [tex]\( g(0) = 4 \)[/tex]. Hence, option A is not necessarily true.
B. horizontal asymptote of [tex]$y=4$[/tex]:
- Horizontal asymptotes are related to the values that a function approaches as [tex]\( x \)[/tex] tends to [tex]\(\pm\infty\)[/tex]. The transformation [tex]\( x \rightarrow x + 4 \)[/tex] in [tex]\( f(x + 4) \)[/tex] does not affect the horizontal asymptote. If [tex]\( f(x) \)[/tex] has a horizontal asymptote of [tex]\( y = L \)[/tex], then [tex]\( g(x) \)[/tex] also has the same horizontal asymptote [tex]\( y = L \)[/tex]. Hence, this option cannot be determined without knowing the behavior of [tex]\( f(x) \)[/tex].
C. [tex]$x$[/tex]-intercept at [tex]$(4, 0)$[/tex]:
- The [tex]$x$[/tex]-intercept for [tex]\( g(x) = f(x + 4) \)[/tex] occurs where [tex]\( g(x) = 0 \)[/tex]. If [tex]\( g(a) = 0 \)[/tex], then
[tex]\[
f(a + 4) = 0.
\][/tex]
Setting [tex]\( b = a + 4 \Rightarrow b = 4 \)[/tex], [tex]\( f(4) = 0 \)[/tex]. No such information allows affirmatively stating that the intercept occurs at [tex]$(4, 0)$[/tex]. Hence, this option is generally not applicable without the specific intercepts of [tex]\( f(x) \)[/tex].
D. horizontal asymptote of [tex]$y=0$[/tex]:
- As previously noted, the horizontal transformation [tex]\( f(x + 4) \)[/tex] does not affect the asymptote of the function. Thus, if the original [tex]\( f(x) \)[/tex] has a horizontal asymptote of [tex]\( y = 0 \)[/tex], the function [tex]\( g(x) = f(x + 4) \)[/tex] will also have the same asymptote,
[tex]\[
\text{A horizontal asymptote at } y = 0.
\][/tex]
Thus, the correct statement describing a key feature of the function [tex]\( g(x) = f(x + 4) \)[/tex] is option D: there is a horizontal asymptote of [tex]\( y = 0 \)[/tex].
