Exponential Functions: Properties, Graphs, and Applications

Exponential Functions

Definition and Properties of Exponents

Exponential functions are a fundamental concept in precalculus, describing functions where the variable appears in the exponent. Understanding their properties is essential for analyzing their behavior and applications.

• Definition: An exponential function with base a is defined as $f(x) = a^x$, where $a > 0$ and $a eq 1$.

• Key Properties of Exponents:

  • (a) $a^x$ is a unique real number for each real number $x$.

  • (b) $a^b = a^c$ if and only if $b = c$.

  • (c) If $a > 1$, then $m < n$ if and only if $a^m < a^n$.

  • (d) If $0 < a < 1$, then $m < n$ if and only if $a^m > a^n$.

Graphs of Exponential Functions

The graph of an exponential function varies depending on the value of the base a. The domain of all exponential functions is $(- , )$, and the range is $(0, )$.

Case 1: $a > 1$

• Example: $f(x) = 2^x$

• Domain: $(- , )$

• Range: $(0, )$

• Behavior: The function increases rapidly as $x$ increases.

• Horizontal Asymptote: The x-axis ($y = 0$) is a horizontal asymptote.

• Key Points: $(0, 1)$, $(1, 2)$, $(-1, 0.5)$

Case 2: $0 < a < 1$

• Example: $g(x) = igg( rac{1}{2}igg)^x$

• Domain: $(- , )$

• Range: $(0, )$

• Behavior: The function decreases as $x$ increases.

• Horizontal Asymptote: The x-axis ($y = 0$) is a horizontal asymptote.

• Key Points: $(0, 1)$, $(1, 0.5)$, $(-1, 2)$

Table of Values for Exponential Functions

Tables are useful for plotting exponential functions and observing their growth or decay.

X	$f(x) = 2^x$	$g(x) = (1/2)^x$
-4	0.0625	16
-2	0.25	4
-1	0.5	2
0	1	1
1	2	0.5
2	4	0.25
3	8	0.125
4	16	0.0625

Additional info: The table above is inferred from the provided images and typical exponential function behavior.

Summary of Exponential Function Characteristics

• Increasing vs. Decreasing:

  • If $a > 1$, $f(x) = a^x$ is increasing.

  • If $0 < a < 1$, $f(x) = a^x$ is decreasing.

• Continuity: Exponential functions are continuous for all real $x$.

• One-to-One: Exponential functions are one-to-one, meaning each $x$ value maps to a unique $y$ value.

• Intercept: All exponential functions pass through $(0, 1)$.

Applications of Exponential Functions

• Growth and Decay: Exponential functions model population growth, radioactive decay, and compound interest.

• Compound Interest: The formula $A = P(1 + r/n)^{nt}$ is used to calculate compound interest, where $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of compounding periods per year, and $t$ is the number of years.

Example: Graphing $f(x) = 2^x$ and $g(x) = (1/2)^x$

• For $f(x) = 2^x$, the graph rises steeply to the right and approaches zero to the left.

• For $g(x) = (1/2)^x$, the graph falls steeply to the right and rises to the left.

Conclusion: Exponential functions are essential in modeling real-world phenomena and understanding their properties is crucial for further study in mathematics and science.