To determine how the graph of [tex]\( g(x) \)[/tex] compares to the graph of [tex]\( f(x) \)[/tex], let's closely analyze the forms of the two functions:
[tex]\[
f(x) = 6^x - 2
\][/tex]
[tex]\[
g(x) = 0.4 \cdot 6^x - 2
\][/tex]
Both functions have the same base value for the exponential term [tex]\( 6^x \)[/tex], and both functions have a constant term subtracted by 2. The primary difference lies in the coefficient of the exponential term.
For [tex]\( f(x) = 6^x - 2 \)[/tex], the coefficient is 1.
For [tex]\( g(x) = 0.4 \cdot 6^x - 2 \)[/tex], the coefficient is 0.4, which is less than 1. When a function of the form [tex]\( a \cdot f(x) \)[/tex] is considered with [tex]\( 0 < a < 1 \)[/tex], this indicates a vertical compression of the original graph by a factor of [tex]\( a \)[/tex]. In this case, [tex]\( a = 0.4 \)[/tex].
This means that the graph of [tex]\( g(x) \)[/tex] is a vertically compressed version of the graph of [tex]\( f(x) \)[/tex], and they both will follow the same general shape but with [tex]\( g(x) \)[/tex] being "squeezed" closer to the x-axis by a factor of 0.4 compared to [tex]\( f(x) \)[/tex].
Therefore, the correct answer is:
C. The graph of [tex]\( g \)[/tex] is a vertical compression of the graph of [tex]\( f \)[/tex].
