To determine which of the given options correctly represents the function [tex]\( g(x) \)[/tex] as [tex]\( f(x) \)[/tex] after shifting to the left by 4 units and up by 6 units, we need to understand how function transformations work.
1. Shifting to the left by 4 units:
A leftward shift of a function [tex]\( f(x) \)[/tex] by 4 units is achieved by replacing [tex]\( x \)[/tex] with [tex]\( x + 4 \)[/tex]. Thus, [tex]\( f(x) \)[/tex] becomes [tex]\( f(x + 4) \)[/tex].
2. Shifting up by 6 units:
An upward shift of a function [tex]\( f(x) \)[/tex] by 6 units is achieved by adding 6 to the entire function. Thus, [tex]\( f(x) \)[/tex] becomes [tex]\( f(x) + 6 \)[/tex].
Combining these two transformations, we first shift [tex]\( f(x) \)[/tex] to the left by 4 units, resulting in [tex]\( f(x + 4) \)[/tex]. Then, we shift this new function [tex]\( f(x + 4) \)[/tex] up by 6 units, which results in [tex]\( f(x + 4) + 6 \)[/tex].
Thus, the expression for [tex]\( g(x) \)[/tex] should be:
[tex]\[ g(x) = f(x + 4) + 6 \][/tex]
Hence, the correct choice is:
[tex]\[ g(x) = f(x+4) + 6 \][/tex]
This matches the fourth option in the given list. So, the answer is:
[tex]\[ 4 \][/tex]
