Let's analyze the given function and statements step-by-step to identify the correct answers.
1. Function Analysis:
- We are given a function [tex]\( f(e) = 24 \)[/tex]. This means that [tex]\( f \)[/tex] is a constant function, where for any input [tex]\( e \)[/tex], the output is always 24.
- The function [tex]\( g(x) \)[/tex] is defined as [tex]\( g(x) = f(x + 2) \)[/tex]. Since [tex]\( f \)[/tex] is a constant function, [tex]\( g(x) \)[/tex] will also be constant regardless of the transformation. Specifically, [tex]\( g(x) = 24 \)[/tex] for all [tex]\( x \)[/tex].
2. Key Features of [tex]\( g(x) \)[/tex]:
- Horizontal Asymptote: Since [tex]\( g(x) = 24 \)[/tex] is a constant function, it does not have a horizontal asymptote at [tex]\( y = 2 \)[/tex] or any other value except [tex]\( y = 24 \)[/tex]. Therefore, statements mentioning horizontal asymptotes at [tex]\( y = 2 \)[/tex] or [tex]\( y = 0 \)[/tex] are not correct.
- Y-intercept: To find the y-intercept of [tex]\( g(x) \)[/tex], we evaluate [tex]\( g(0) \)[/tex]. Given [tex]\( g(x) = 24 \)[/tex], we have [tex]\( g(0) = 24 \)[/tex]. This gives us the y-intercept at the point [tex]\( (0, 24) \)[/tex]. Therefore, none of the [tex]\( y \)[/tex]-intercept options provided ([tex]\( (0, 1) \)[/tex] or [tex]\( (0, 4) \)[/tex]) are correct.
- Domain: The domain of [tex]\( g(x) = 24 \)[/tex] is all real numbers, as there are no restrictions on [tex]\( x \)[/tex]. This is true for any constant function.
3. Summary:
- Correct statement: The domain of [tex]\( g(x) \)[/tex] is [tex]\( \{ x \mid -\infty < x < \infty \} \)[/tex].
- Incorrect statements:
- Horizontal asymptote of [tex]\( y = 2 \)[/tex]
- Y-intercept at [tex]\( (0, 1) \)[/tex]
- Horizontal asymptote of [tex]\( y = 0 \)[/tex]
- Y-intercept at [tex]\( (0, 4) \)[/tex]
Thus, the correct answer from the provided options is:
- Domain of [tex]\( \{ x \mid -\infty < x < \infty \} \)[/tex].
These are the only key features that correctly describe the function [tex]\( g(x) = 24 \)[/tex].
