Question

the equations of two exponential functions are shown.
f(x) = 9(3/8)^x
g(x) = 9(8/3)^x
which two statements are true about the graphs of the functions

Answer 1

To analyze the two exponential functions provided and determine the true statements about their graphs, let’s break down the functions:

1. **Function f(x) = 9(3/8)^x**:
- This function has a base of \( \frac{3}{8} \), which is a fraction less than 1, indicating that it is a **decaying exponential**. As \( x \) increases, \( f(x) \) approaches 0.

2. **Function g(x) = 9(8/3)^x**:
- This function has a base of \( \frac{8}{3} \), which is greater than 1, indicating that it is a **growing exponential**. As \( x \) increases, \( g(x) \) increases rapidly towards infinity.

### True Statements About the Graphs:
Given the characteristics of the two functions, here are two statements that can be considered true:

1. **As \( x \to \infty \), f(x) approaches 0, while g(x) approaches infinity**:
- This is because f(x) is a decaying function (with a base less than 1) and g(x) is a growing function (with a base greater than 1).

2. **The graphs of f(x) and g(x) will intersect at a point in the positive x-axis**:
- There exists a point where the decaying function f(x) and the growing function g(x) are equal, meaning they will intersect at least once for some value of \( x \).

### Additional Observations:
- Both functions are defined for all real numbers.
- Both functions are positive for all \( x \) because they multiply by a positive coefficient (9).

In summary, the two true statements about the graphs of the functions are:
1. \( f(x) \) is decreasing and approaches 0 as \( x \) increases, while \( g(x) \) is increasing and approaches infinity as \( x \) increases.
2. The two functions will intersect at least once somewhere in their domain.

If you have specific statements that you would like to verify or further explore, please provide them!