To determine whether the function [tex]\( f(x) = 5 e^{0.25 x} \)[/tex] represents exponential growth, decay, or neither, let's analyze its form and the properties of exponential functions.
An exponential function is generally of the form:
[tex]\[ f(x) = a e^{bx} \][/tex]
Where:
- [tex]\( a \)[/tex] is a constant.
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately 2.71828).
- [tex]\( b \)[/tex] is the exponent, which is a constant.
For the given function [tex]\( f(x) = 5 e^{0.25 x} \)[/tex]:
- [tex]\( a \)[/tex] is 5.
- [tex]\( b \)[/tex] is 0.25.
1. Exponential Growth or Decay:
- If [tex]\( b \)[/tex] is positive ([tex]\( b > 0 \)[/tex]), the function represents exponential growth.
- If [tex]\( b \)[/tex] is negative ([tex]\( b < 0 \)[/tex]), the function represents exponential decay.
Here, [tex]\( b \)[/tex] (the exponent) is 0.25, which is a positive number.
Since the exponent 0.25 is positive, we conclude that the function [tex]\( f(x) = 5 e^{0.25 x} \)[/tex] represents exponential growth.
Hence, the correct answer is:
D) Exponential growth
