Certainly! Let's examine this problem step-by-step:
### Understanding the Problem
Given:
[tex]\[ g(x) = f(x + 2) \][/tex]
We have a function [tex]\( g \)[/tex] that is derived from another function [tex]\( f \)[/tex] by applying a horizontal shift of 2 units to the left. To find the equation of the asymptote of [tex]\( g \)[/tex], we need to consider the asymptote of the original function [tex]\( f \)[/tex].
### Asymptotes and Horizontal Shifts
1. Asymptote Basics: An asymptote is a line that a graph of a function approaches but never touches. For the function [tex]\( f(x) \)[/tex], let's denote its asymptote by the equation [tex]\( y = a \)[/tex].
2. Effect of Horizontal Shift: When you apply a horizontal shift to a function, which in this case is [tex]\( f(x) \to f(x + 2) \)[/tex], the asymptote does not change. This is because horizontal shifts do not affect the [tex]\( y \)[/tex]-values of the function, only the [tex]\( x \)[/tex]-values.
### Steps
1. Identify the asymptote of [tex]\( f(x) \)[/tex]:
- Assume the asymptote of [tex]\( f(x) \)[/tex] is [tex]\( y = a \)[/tex].
2. Determine the asymptote of [tex]\( g(x) \)[/tex]:
- Since [tex]\( g(x) = f(x + 2) \)[/tex] is simply a horizontally shifted version of [tex]\( f(x) \)[/tex], the asymptote of [tex]\( g \)[/tex] remains the same as that of [tex]\( f \)[/tex].
- Hence, the asymptote of [tex]\( g(x) \)[/tex] is also [tex]\( y = a \)[/tex].
### Final Answer
The equation of the asymptote for the function [tex]\( g(x) = f(x + 2) \)[/tex] is:
[tex]\[ y = a \][/tex]
Thus, horizontal shifts do not alter the asymptote of a function.
