A function is shown:
[tex]f(x)=(1.07)^x[/tex]
What does the function represent?
A. Exponential growth of 7%
B. Exponential decay of 7%
C. Exponential growth of 93%
D. Exponential decay of 93%
Answer (1)
The function is in the form f ( x ) = a x with a = 1.07 .
Since 1"> a > 1 , it represents exponential growth.
The percentage growth is calculated as ( 1.07 − 1 ) × 100% = 7% .
Therefore, the function represents exponential growth of 7% .
Explanation
Understanding the Function We are given the function f ( x ) = ( 1.07 ) x and asked to determine what it represents.
Identifying the Form The function is in the form of an exponential function, f ( x ) = a x , where a is a constant. In this case, a = 1.07 .
Determining Growth or Decay If 1"> a > 1 , the function represents exponential growth. If 0 < a < 1 , the function represents exponential decay. Since 1"> 1.07 > 1 , the function represents exponential growth.
Calculating Percentage Growth To find the percentage growth, we calculate ( a − 1 ) × 100% . In this case, ( 1.07 − 1 ) × 100% = 0.07 × 100% = 7% .
Conclusion Therefore, the function f ( x ) = ( 1.07 ) x represents exponential growth of 7%.
Examples
Imagine you invest money in a bank account that offers an annual interest rate of 7%, compounded annually. The function f ( x ) = ( 1.07 ) x models how your investment grows over time, where x is the number of years. This function shows that each year, your investment increases by 7% of its previous value. Understanding exponential growth is crucial for making informed financial decisions, such as planning for retirement or understanding the impact of inflation on your savings.
