To analyze the two exponential functions provided and determine the true statements about their graphs, let’s break down the functions:
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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.
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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:
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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).
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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:
- \( f(x) \) is decreasing and approaches 0 as \( x \) increases, while \( g(x) \) is increasing and approaches infinity as \( x \) increases.
- 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!