One of the most common examples of an exponential function is population growth. When a population grows at a constant percentage rate, it can be modeled using the exponential growth formula:[ P(t) = P_0 e^{rt} ]where:

• ( P(t) ) is the population at time ( t ),
• ( P_0 ) is the initial population,
• ( r ) is the growth rate,
• ( e ) is Euler's number (approximately 2.718).For instance, if a rabbit population starts with 100 rabbits and grows at a rate of 20% per year, the population after five years can be calculated using this formula.

Another significant example is radioactive decay, where the quantity of a radioactive substance decreases over time at a rate proportional to its current amount. The decay can be modeled by the formula:[ N(t) = N_0 e^{-\lambda t} ]where:

• ( N(t) ) is the remaining quantity at time ( t ),
• ( N_0 ) is the initial quantity,
• ( \lambda ) is the decay constant.This formula is used in fields like nuclear physics and archaeology (carbon dating) to determine the age of artifacts.

In finance, compound interest is another classic example of an exponential function. The amount of money accumulated over time can be calculated using the formula:[ A = P(1 + r/n)^{nt} ]where:

• ( A ) is the amount of money accumulated after n years, including interest,
• ( P ) is the principal amount (the initial sum of money),
• ( r ) is the annual interest rate (decimal),
• ( n ) is the number of times that interest is compounded per year,
• ( t ) is the number of years the money is invested or borrowed.This formula illustrates how investments grow exponentially over time due to the effect of compounding.

In biology, the growth of bacteria can also be modeled using exponential functions. For example, if a single bacterium doubles every hour, the number of bacteria after ( t ) hours can be expressed as:[ N(t) = N_0 \cdot 2^t ]where ( N_0 ) is the initial number of bacteria. This illustrates how quickly populations can grow under ideal conditions.These examples highlight the importance of exponential functions in modeling real-world phenomena across various disciplines, demonstrating their relevance and application in understanding growth and decay processes.

An exponential function is typically expressed in the form (f(x) = a \cdot b^x), where:

• (a) is a constant (the coefficient),
• (b) is the base (a positive real number not equal to 1),
• (x) is the exponent (usually a variable).

1. Growth and Decay:
   • If (b > 1), the function represents exponential growth, increasing rapidly as (x) increases.
   • If (0 < b < 1), the function represents exponential decay, decreasing rapidly as (x) increases.
2. Domain and Range:
   • The domain of an exponential function is all real numbers, ((-∞, ∞)).
   • The range is always positive real numbers, ((0, ∞)), meaning the function never touches the x-axis.
3. Horizontal Asymptote:
   • The line (y = 0) (the x-axis) is a horizontal asymptote for exponential functions, meaning the graph approaches this line but never actually reaches it.
4. Monotonicity:
   • Exponential functions are either increasing or decreasing. They are increasing if (b > 1) and decreasing if (0 < b < 1).
5. Intercepts:
   • All exponential functions pass through the point ((0, a)), where (a) is the coefficient. This is because (f(0) = a \cdot b^0 = a).
6. Graphical Representation:
   • The graph of an exponential function is smooth and continuous, showing a rapid increase or decrease depending on the base. It always passes through the point ((0, 1)) when (a = 1).

Exponential functions are widely used in various fields, including:

• Population growth: Modeling how populations increase over time.
• Finance: Calculating compound interest and investment growth.
• Natural phenomena: Describing processes like radioactive decay and bacterial growth.These properties make exponential functions essential in mathematics and its applications across different disciplines.