Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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].
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].
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
