Answer :
To identify the type of function represented by [tex]\( f(x) = 4 \cdot 2^x \)[/tex], let's analyze the function step-by-step.
1. Form of the Function:
- The given function is [tex]\( f(x) = 4 \cdot 2^x \)[/tex].
- This is in the form of [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( x \)[/tex] is the variable.
2. Identifying Exponential Functions:
- Functions of the form [tex]\( f(x) = a \cdot b^x \)[/tex] are classified as exponential functions.
- If [tex]\( b > 1 \)[/tex], the function represents exponential growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents exponential decay.
- In our function, [tex]\( a = 4 \)[/tex] and [tex]\( b = 2 \)[/tex].
3. Analyzing the Base (b):
- Here, [tex]\( b = 2 \)[/tex].
- Since [tex]\( 2 > 1 \)[/tex], the function [tex]\( f(x) = 4 \cdot 2^x \)[/tex] represents exponential growth.
4. Conclusion:
- Based on the condition that [tex]\( b > 1 \)[/tex], the function [tex]\( f(x) = 4 \cdot 2^x \)[/tex] must be classified as an exponential growth function.
Therefore, the correct answer is:
D. Exponential growth
1. Form of the Function:
- The given function is [tex]\( f(x) = 4 \cdot 2^x \)[/tex].
- This is in the form of [tex]\( f(x) = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( x \)[/tex] is the variable.
2. Identifying Exponential Functions:
- Functions of the form [tex]\( f(x) = a \cdot b^x \)[/tex] are classified as exponential functions.
- If [tex]\( b > 1 \)[/tex], the function represents exponential growth.
- If [tex]\( 0 < b < 1 \)[/tex], the function represents exponential decay.
- In our function, [tex]\( a = 4 \)[/tex] and [tex]\( b = 2 \)[/tex].
3. Analyzing the Base (b):
- Here, [tex]\( b = 2 \)[/tex].
- Since [tex]\( 2 > 1 \)[/tex], the function [tex]\( f(x) = 4 \cdot 2^x \)[/tex] represents exponential growth.
4. Conclusion:
- Based on the condition that [tex]\( b > 1 \)[/tex], the function [tex]\( f(x) = 4 \cdot 2^x \)[/tex] must be classified as an exponential growth function.
Therefore, the correct answer is:
D. Exponential growth
