To determine the transformed function [tex]\( g(x) \)[/tex], we need to apply a horizontal compression by a factor of [tex]\( \frac{1}{3} \)[/tex] to the original function [tex]\( f(x) = 7^x + 1 \)[/tex].
Horizontal compression by a factor of [tex]\( \frac{1}{3} \)[/tex] involves multiplying the input variable [tex]\( x \)[/tex] by 3. Therefore, if the original function is [tex]\( f(x) = 7^x + 1 \)[/tex], then the horizontally compressed function [tex]\( g(x) \)[/tex] can be represented as:
[tex]\[ g(x) = f(3x) \][/tex]
Substituting [tex]\( 3x \)[/tex] into the original function:
[tex]\[ g(x) = 7^{3x} + 1 \][/tex]
Thus, the transformed function [tex]\( g(x) \)[/tex] is given by:
[tex]\[ g(x) = 7^{3x} + 1 \][/tex]
This is the equation of the function [tex]\( g \)[/tex] after the horizontal compression.
