Exponential Functions: Definitions, Properties, and Graphs

Exponential Functions

Introduction to Exponential Functions

Exponential functions are mathematical models where a constant base is raised to a variable exponent. They are widely used to describe growth and decay processes in fields such as finance, biology, and physics.

• Exponential growth occurs when the base is greater than 1.

• Exponential decay occurs when the base is between 0 and 1.

Comparing Growth Options: Doubling vs. Fixed Amount

Consider two options for receiving money:

• Option a: $1 million dollars right now.

• Option b: $1 that doubles every day for 1 month.

Calculating the total for option b over several days demonstrates the rapid increase due to exponential growth:

• Day 1: $1

• Day 2: $2

• Day 3: $4

• Day 4: $8

• Day 5: $16

• Day 6: $32

• Day 7: $64

• Day 8: $128

• Day 9: $256

• Day 10: $512

• Day 11: $1,024

• Day 12: $2,048

• Day 13: $4,096

• Day 14: $8,192

• Day 15: $16,384

• Day 16: $32,768

• Day 17: $65,536

• Day 18: $131,072

• Day 19: $262,144

• Day 20: $524,288

• Day 21: $1,048,576

• Day 22: $2,097,152

• Day 23: $4,194,304

• Day 24: $8,388,608

• Day 25: $16,777,216

• Day 26: $33,554,432

• Day 27: $67,108,864

• Day 28: $134,217,728

• Day 29: $268,435,456

• Day 30: $536,870,912

Conclusion: Option b, which doubles each day, far exceeds the fixed $1 million after 30 days, illustrating the power of exponential growth.

Formulas for Exponential Growth

• General exponential function: $f(x) = C a^x$

• Growth factor: $a$ (if $a > 1$, growth; if $0 < a < 1$, decay)

• Initial value: $C$

Example of recursive calculation:

• $P(1) = 0.01$

• $P(2) = 2 \cdot P(1) = 2 \cdot 0.01$

• $P(3) = 2 \cdot P(2) = 2 \cdot (2 \cdot 0.01)$

• $P(4) = 2 \cdot P(3) = 2 \cdot (2 \cdot (2 \cdot 0.01)) = 8 \cdot 0.01$

• General form: $P(x) = 2^{x-1} \cdot 0.01$

Definition: Exponential Function

• An exponential function is any function of the form $f(x) = C a^x$, where $a > 0$ and $a \neq 1$.

• Growth factor: $a$

• Initial value: $C$

Properties of Exponential Functions

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Monotonicity: Always increasing if $a > 1$, always decreasing if $0 < a < 1$

• Horizontal asymptote: $y = 0$

Behavior Based on the Base

• If $a > 1$, $f(x) = a^x$ is increasing with domain $(-\infty, \infty)$ and range $(0, \infty)$.

• If $0 < a < 1$, $f(x) = a^x$ is decreasing with domain $(-\infty, \infty)$ and range $(0, \infty)$.

Graphing Exponential Functions

To graph exponential functions, create a table of values and plot the points. For example:

x	$f(x) = 2^x$
-2	1/4
-1	1/2
0	1
1	2
2	4
3	8

x	$f(x) = (1/2)^x$
-2	4
-1	2
0	1
1	1/2
2	1/4
3	1/8

These tables illustrate the rapid increase or decrease characteristic of exponential functions.

Graphical Representation

The graphs of $f(x) = 2^x$ and $f(x) = (1/2)^x$ show:

• $f(x) = 2^x$ rises steeply as $x$ increases and approaches zero as $x$ decreases.

• $f(x) = (1/2)^x$ falls as $x$ increases and rises as $x$ decreases, but never touches the x-axis.

Horizontal asymptote: Both functions approach $y = 0$ as $x \to -\infty$ (for $a > 1$) or $x \to \infty$ (for $0 < a < 1$).

Summary Table: Properties of Exponential Functions

Property	$a > 1$	$0 < a < 1$
Monotonicity	Increasing	Decreasing
Domain	$(-\infty, \infty)$
Range	$(0, \infty)$
Horizontal Asymptote	$y = 0$