Complete the input-output table for the function [tex]y=3^x[/tex].
| x | y |
| --- | ----- |
| -2 | 1 / 9 |
| -1 | 1 / 3 |
| 0 | 1 |
| 1 | 3 |
| 2 | 9 |
| 3 | a |
| 4 | b |
[tex]a=[/tex] [tex]b=[/tex]
Answer (2)
Substitute x = 3 into the function y = 3 x to find a : a = 3 3 = 27 .
Substitute x = 4 into the function y = 3 x to find b : b = 3 4 = 81 .
The value of a is 27 and the value of b is 81.
Therefore, the final answer is a = 27 , b = 81 .
Explanation
Understanding the problem We are given the function y = 3 x and a table with x and y values. We need to find the values of a and b where a is the value of y when x = 3 and b is the value of y when x = 4 .
Calculating a To find the value of a , we substitute x = 3 into the function y = 3 x . Thus, a = 3 3 .
Finding the value of a We calculate 3 3 as 3 × 3 × 3 = 9 × 3 = 27 . Therefore, a = 27 .
Calculating b To find the value of b , we substitute x = 4 into the function y = 3 x . Thus, b = 3 4 .
Finding the value of b We calculate 3 4 as 3 × 3 × 3 × 3 = 9 × 9 = 81 . Therefore, b = 81 .
Final Answer Therefore, a = 27 and b = 81 .
Examples
Exponential functions like y = 3 x are crucial in modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For instance, imagine a bacterial colony that triples in size every hour. If you start with one bacterium, after x hours, you'll have 3 x bacteria. This table helps you quickly see how the colony grows: after 3 hours, you'd have 27 bacteria, and after 4 hours, you'd have 81. Understanding exponential growth allows us to predict and manage these processes effectively.
To complete the table for y = 3 x , we find a = 27 and b = 81 by calculating 3 3 and 3 4 respectively. Therefore, the results are a = 27 and b = 81 .
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