To determine which of the given options matches [tex]\( g(x) \)[/tex] based on the function [tex]\( f(x) = 4^x \)[/tex], let's evaluate each of the options in terms of [tex]\( f(x) \)[/tex].
Given [tex]\( f(x) = 4^x \)[/tex], we want to find the correct form for [tex]\( g(x) \)[/tex]:
### Option A: [tex]\( 4^x - 2 \)[/tex]
This option is expressed as [tex]\( f(x) - 2 \)[/tex], or:
[tex]\[ g(x) = 4^x - 2 \][/tex]
This is not an exponential function of the same form because subtracting 2 is an arithmetic operation, not an exponential transformation.
### Option B: [tex]\( 4^x + 2 \)[/tex]
This option is expressed as [tex]\( f(x) + 2 \)[/tex], or:
[tex]\[ g(x) = 4^x + 2 \][/tex]
Similarly, this is not an exponential function of the same form because adding 2 is an arithmetic operation, not an exponential transformation.
### Option C: [tex]\( 4^{(x+2)} \)[/tex]
This option applies an exponent rule of [tex]\( f(x + 2) \)[/tex]. Let's rewrite it:
[tex]\[ g(x) = 4^{(x+2)} \][/tex]
Using the properties of exponents, we can rewrite this as:
[tex]\[ g(x) = 4^x \cdot 4^2 \][/tex]
[tex]\[ g(x) = 4^x \cdot 16 \][/tex]
This is an appropriate transformation because it retains the exponential format and just scales the base function [tex]\( f(x) \)[/tex].
### Option D: [tex]\( 4^{(x-2)} \)[/tex]
This option applies an exponent rule of [tex]\( f(x - 2) \)[/tex]. Let's rewrite it:
[tex]\[ g(x) = 4^{(x-2)} \][/tex]
Using the properties of exponents, we can rewrite this as:
[tex]\[ g(x) = 4^x \cdot 4^{-2} \][/tex]
[tex]\[ g(x) = 4^x \cdot \frac{1}{16} \][/tex]
This is another appropriate transformation because it retains the exponential format while scaling the base function [tex]\( f(x) \)[/tex].
From the analysis, Option C matches the required transformation form of [tex]\( g(x) \)[/tex] with [tex]\( f(x) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{4^{(x+2)}} \][/tex]
Which corresponds to:
[tex]\[
\boxed{C}
\][/tex]
