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9. Find the exponential function that represents the data in the table. f(x) 2. 1 2. 18 3. 54 4 162 O fG)-26 O Not an exponential function O f)=2-3* O f)=3 2*

9. Find the exponential function that represents the data in the table. f(x) 2. 1 2. 18 3. 54 4 162 O fG)-26 O Not an exponential function O f)=2-3* O f)=3 2*

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Problem 9: Exponential Function Identification** **Question:** Find the exponential function that represents the data in the table below. **Data Table:** | x | f(x) | |---|------| | 0 | 2 | | 1 | 6 | | 2 | 18 | | 3 | 54 | | 4 | 162 | **Options:** - \( f(x) = 2 \cdot 6^x \) - Not an exponential function - \( f(x) = 2 \cdot 3^x \) - \( f(x) = 3 \cdot 2^x \) **Solution Approach:** To determine the correct exponential function, look for a pattern in the values of \( f(x) \) relative to \( x \). Expanding upon sequences, you can assess if there is a common ratio (multiplier) between subsequent values of \( f(x) \). **Analysis of Data:** - \( x = 0, f(x) = 2 \) - \( x = 1, f(x) = 6 \) - Factor from 2 to 6 is 3. - \( x = 2, f(x) = 18 \) - Factor from 6 to 18 is 3. - \( x = 3, f(x) = 54 \) - Factor from 18 to 54 is 3. - \( x = 4, f(x) = 162 \) - Factor from 54 to 162 is 3. Thus, each \( f(x) \) value is derived by multiplying the previous value by 3. Therefore, the exponential function is \( f(x) = 2 \cdot 3^x \), as confirmed by the initial value at \( x = 0 \). **Correct answer:** - \( f(x) = 2 \cdot 3^x \)
Transcribed Image Text:**Problem 9: Exponential Function Identification** **Question:** Find the exponential function that represents the data in the table below. **Data Table:** | x | f(x) | |---|------| | 0 | 2 | | 1 | 6 | | 2 | 18 | | 3 | 54 | | 4 | 162 | **Options:** - \( f(x) = 2 \cdot 6^x \) - Not an exponential function - \( f(x) = 2 \cdot 3^x \) - \( f(x) = 3 \cdot 2^x \) **Solution Approach:** To determine the correct exponential function, look for a pattern in the values of \( f(x) \) relative to \( x \). Expanding upon sequences, you can assess if there is a common ratio (multiplier) between subsequent values of \( f(x) \). **Analysis of Data:** - \( x = 0, f(x) = 2 \) - \( x = 1, f(x) = 6 \) - Factor from 2 to 6 is 3. - \( x = 2, f(x) = 18 \) - Factor from 6 to 18 is 3. - \( x = 3, f(x) = 54 \) - Factor from 18 to 54 is 3. - \( x = 4, f(x) = 162 \) - Factor from 54 to 162 is 3. Thus, each \( f(x) \) value is derived by multiplying the previous value by 3. Therefore, the exponential function is \( f(x) = 2 \cdot 3^x \), as confirmed by the initial value at \( x = 0 \). **Correct answer:** - \( f(x) = 2 \cdot 3^x \)
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