Complete the table of values for the function.
[tex]f(x)=\left(\frac{1}{3}\right)^x[/tex]
a =
b =
c =
| x | f(x) |
|---|---|
| -2 | a |
| -1 | b |
| 0 | c |
| 1 | 1/3 |
| 2 | 1/9 |
Answer (2)
Calculate a = f ( − 2 ) = ( 3 1 ) − 2 = 9 .
Calculate b = f ( − 1 ) = ( 3 1 ) − 1 = 3 .
Calculate c = f ( 0 ) = ( 3 1 ) 0 = 1 .
The completed table values are a = 9 , b = 3 , and c = 1 , so the final answer is a = 9 , b = 3 , c = 1 .
Explanation
Understanding the Problem We are given the function f ( x ) = ( 3 1 ) x and asked to complete the table of values for x = − 2 , − 1 , 0 . This means we need to find f ( − 2 ) , f ( − 1 ) , and f ( 0 ) .
Calculating a Let's calculate a = f ( − 2 ) . We have f ( − 2 ) = ( 3 1 ) − 2 . Recall that a − n = a n 1 , so ( 3 1 ) − 2 = ( 3 1 ) 2 1 = 9 1 1 = 9 . Therefore, a = 9 .
Calculating b Next, let's calculate b = f ( − 1 ) . We have f ( − 1 ) = ( 3 1 ) − 1 . Using the same property as above, ( 3 1 ) − 1 = 3 1 1 = 3 . Therefore, b = 3 .
Calculating c Finally, let's calculate c = f ( 0 ) . We have f ( 0 ) = ( 3 1 ) 0 . Recall that any non-zero number raised to the power of 0 is 1. Therefore, c = 1 .
Final Answer Thus, we have a = 9 , b = 3 , and c = 1 .
Examples
Exponential functions like f ( x ) = ( 3 1 ) x are used to model various real-world phenomena, such as radioactive decay or the depreciation of an asset. For example, if you invest in a car, its value decreases over time. If the value decreases by a third each year, the car's value after x years can be modeled by a function similar to the one in this problem. Understanding how to evaluate such functions helps in predicting future values and making informed decisions.
To complete the table, we find that a = 9 , b = 3 , and c = 1 for the function f ( x ) = ( 3 1 ) x . The calculations come from the properties of exponents. The final values for the table are a = 9 , b = 3 , and c = 1 .
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