To analyze the function \( f(x) = 3(9)^x \), we can rewrite \( 9 \) in terms of a base of \( 3 \):
\[ 9 = 3^2 \]
Thus, we can express \( (9)^x \) as:
\[ (9)^x = (3^2)^x = 3^{2x} \]
Now, we can rewrite the function \( f(x) \):
\[ f(x) = 3(9)^x = 3(3^{2x}) \]
Using the laws of exponents, we can combine the terms:
\[ f(x) = 3^{1 + 2x} \]
Now, the function \( f(x) \) can be expressed as an exponential function with base \( 3 \):
\[ f(x) = 3^{2x + 1} \]
From this representation, we can determine several properties of the function:
- Growth: The function is an exponential function, which means it will grow rapidly as \( x \) increases.
- Y-Intercept: When \( x = 0 \), \( f(0) = 3^{2(0) + 1} = 3^1 = 3 \).
- Behavior: As \( x \to -\infty \), \( f(x) \to 0 \); as \( x \to \infty \), \( f(x) \to \infty \).
- Monotonicity: The function is increasing for all \( x \) since the base \( 3 \) is greater than \( 1 \).
Given this analysis, any of the statements regarding these properties would be true for the function \( f(x) = 3(9)^x \). If you have specific statements you want to evaluate for truthfulness, please provide them!