Question

Which of these statements is true for

`f\left(x\right)\ =3\left(9\right)^{x}\ `

Answer 1

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:

1. **Growth**: The function is an exponential function, which means it will grow rapidly as \( x \) increases.
2. **Y-Intercept**: When \( x = 0 \), \( f(0) = 3^{2(0) + 1} = 3^1 = 3 \).
3. **Behavior**: As \( x \to -\infty \), \( f(x) \to 0 \); as \( x \to \infty \), \( f(x) \to \infty \).
4. **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!