To solve the problem, let's first clarify the given functions and understand their properties step-by-step.
1. Given Function [tex]\( f(x) \)[/tex]:
The function [tex]\( f(x) \)[/tex] is defined as:
[tex]\[
f(x) = \frac{2}{151}
\][/tex]
This is a constant function, meaning that regardless of the value of [tex]\( x \)[/tex], the output of [tex]\( f(x) \)[/tex] is always [tex]\(\frac{2}{151}\)[/tex].
2. Transformed Function [tex]\( g(x) \)[/tex]:
The function [tex]\( g(x) \)[/tex] is defined as:
[tex]\[
g(x) = f(x+2)
\][/tex]
Here, [tex]\( g(x) \)[/tex] is essentially the function [tex]\( f(x) \)[/tex] evaluated at [tex]\( x + 2 \)[/tex]. However, since [tex]\( f(x) \)[/tex] is a constant, shifting the input by 2 units (i.e., [tex]\( x + 2 \)[/tex]) does not affect the output value of the function. Therefore:
[tex]\[
g(x) = f(x+2) = \frac{2}{151}
\][/tex]
This indicates that [tex]\( g(x) \)[/tex] is also a constant function with the same value, [tex]\(\frac{2}{151}\)[/tex], for all [tex]\( x \)[/tex].
3. Asymptotes of a Constant Function:
In the context of asymptotes, since [tex]\( g(x) \)[/tex] is a constant function, it does not have any vertical asymptotes (which occur in functions that approach infinity) nor oblique asymptotes (which are relevant for polynomial functions of degree higher than 1). The only relevant asymptote here is a horizontal asymptote.
4. Horizontal Asymptote:
A horizontal asymptote represents the value that [tex]\( g(x) \)[/tex] approaches as [tex]\( x \)[/tex] goes to positive or negative infinity. For a constant function like [tex]\( g(x) = \frac{2}{151} \)[/tex], the function value remains constant at [tex]\(\frac{2}{151}\)[/tex] regardless of [tex]\( x \)[/tex]. Hence, the function [tex]\( g(x) \)[/tex] itself is the horizontal line [tex]\( y = \frac{2}{151} \)[/tex].
Therefore, the equation of the horizontal asymptote of [tex]\( g(x) \)[/tex] is:
[tex]\[
y = \frac{2}{151}
\][/tex]
Conclusively, the equation of the asymptote of [tex]\( g(x) \)[/tex] is:
[tex]\[
y = 0.013245033112582781
\][/tex]
