Answer :
To identify the type of function represented by [tex]\( f(x) = 2\left(\frac{1}{2}\right)^x \)[/tex], let's analyze its form step by step.
1. Understanding the Function's Form:
The given function [tex]\( f(x) = 2\left(\frac{1}{2}\right)^x \)[/tex] is of the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
2. Identifying the Constants:
In this function, [tex]\( a = 2 \)[/tex] and [tex]\( b = \frac{1}{2} \)[/tex].
3. Property of the Base [tex]\( b \)[/tex]:
- For exponential functions, [tex]\( b \)[/tex] (the base) determines the behavior of the function.
- If [tex]\( b > 1 \)[/tex], the function represents exponential growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents exponential decay.
- In this case, [tex]\( b = \frac{1}{2} \)[/tex], which is less than 1 but greater than 0.
4. Conclusion about the Type of Function:
- Since [tex]\( \frac{1}{2} \)[/tex] (which is [tex]\( b \)[/tex]) is between 0 and 1, the function [tex]\( f(x) = 2\left(\frac{1}{2}\right)^x \)[/tex] represents exponential decay.
Therefore, the type of function represented by [tex]\( f(x) = 2\left(\frac{1}{2}\right)^x \)[/tex] is Exponential decay, corresponding to option B.
1. Understanding the Function's Form:
The given function [tex]\( f(x) = 2\left(\frac{1}{2}\right)^x \)[/tex] is of the form [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
2. Identifying the Constants:
In this function, [tex]\( a = 2 \)[/tex] and [tex]\( b = \frac{1}{2} \)[/tex].
3. Property of the Base [tex]\( b \)[/tex]:
- For exponential functions, [tex]\( b \)[/tex] (the base) determines the behavior of the function.
- If [tex]\( b > 1 \)[/tex], the function represents exponential growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents exponential decay.
- In this case, [tex]\( b = \frac{1}{2} \)[/tex], which is less than 1 but greater than 0.
4. Conclusion about the Type of Function:
- Since [tex]\( \frac{1}{2} \)[/tex] (which is [tex]\( b \)[/tex]) is between 0 and 1, the function [tex]\( f(x) = 2\left(\frac{1}{2}\right)^x \)[/tex] represents exponential decay.
Therefore, the type of function represented by [tex]\( f(x) = 2\left(\frac{1}{2}\right)^x \)[/tex] is Exponential decay, corresponding to option B.
