Answer :
To determine which term best describes the graph of the exponential function [tex]\( f(x) = 9 \cdot \left(\frac{1}{7}\right)^x \)[/tex], we need to analyze the properties of exponential functions.
1. Understanding the Structure of the Function:
- The function is given in the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a = 9 \)[/tex] and [tex]\( b = \frac{1}{7} \)[/tex].
2. Analyzing the Base [tex]\( b \)[/tex]:
- The base [tex]\( b = \frac{1}{7} \)[/tex] is a fraction less than 1 (since [tex]\( \frac{1}{7} \)[/tex] is approximately 0.1428).
- In exponential functions, when [tex]\( b \)[/tex] (the base) is a number between 0 and 1, the function is decreasing.
3. Determining the Behavior:
- For [tex]\( b \in (0, 1) \)[/tex], as [tex]\( x \)[/tex] increases, [tex]\( b^x \)[/tex] decreases. Specifically, [tex]\( \left(\frac{1}{7}\right)^x \)[/tex] decreases as [tex]\( x \)[/tex] becomes larger.
- This implies that [tex]\( f(x) = 9 \cdot \left(\frac{1}{7}\right)^x \)[/tex] is a decreasing function.
4. Conclusion:
- Given the properties of functions with a base between 0 and 1, the term that best describes the behavior of [tex]\( f(x) \)[/tex] is "decreasing."
Therefore, the correct answer is:
D. Decreasing
1. Understanding the Structure of the Function:
- The function is given in the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a = 9 \)[/tex] and [tex]\( b = \frac{1}{7} \)[/tex].
2. Analyzing the Base [tex]\( b \)[/tex]:
- The base [tex]\( b = \frac{1}{7} \)[/tex] is a fraction less than 1 (since [tex]\( \frac{1}{7} \)[/tex] is approximately 0.1428).
- In exponential functions, when [tex]\( b \)[/tex] (the base) is a number between 0 and 1, the function is decreasing.
3. Determining the Behavior:
- For [tex]\( b \in (0, 1) \)[/tex], as [tex]\( x \)[/tex] increases, [tex]\( b^x \)[/tex] decreases. Specifically, [tex]\( \left(\frac{1}{7}\right)^x \)[/tex] decreases as [tex]\( x \)[/tex] becomes larger.
- This implies that [tex]\( f(x) = 9 \cdot \left(\frac{1}{7}\right)^x \)[/tex] is a decreasing function.
4. Conclusion:
- Given the properties of functions with a base between 0 and 1, the term that best describes the behavior of [tex]\( f(x) \)[/tex] is "decreasing."
Therefore, the correct answer is:
D. Decreasing
