Exponential Function

Understanding Exponential Functions

An exponential function is a mathematical function that has a variable as the exponent of a constant number. The general form of an exponential function is $f(x) = a \cdot b^x$ with $a \neq 0$ , $b > 0$ , and $b \neq 1$ .

Components of exponential functions:

In the function $f(x) = a \cdot b^x$ :

$a$ is the multiplier constant that determines the initial value of the function
$b$ is the exponential base that determines the rate of growth or decay
$x$ is the independent variable (exponent)

Characteristics of Exponential Functions

Exponential functions have several special properties that distinguish them from other functions:

Basic Properties

f(0) = a \cdot b^0 = a \cdot 1 = a

f(x_1 + x_2) = a \cdot b^{x_1 + x_2} = a \cdot b^{x_1} \cdot b^{x_2}

f(x_1 - x_2) = a \cdot b^{x_1 - x_2} = \frac{a \cdot b^{x_1}}{b^{x_2}}

Types of Exponential Functions

Exponential Growth Function ( $b > 1$ ):

Function values increase as $x$ increases
Graph rises from left to right
Example: $f(x) = 2^x$

Exponential Decay Function ( $0 < b < 1$ ):

Function values decrease as $x$ increases
Graph falls from left to right
Example: $f(x) = (0.5)^x$

Graphs of Exponential Functions

The following is a visualization of various exponential functions:

Comparison of Exponential Functions

The graph shows the growth function

f(x) = 2^x

and decay function

g(x) = (0.5)^x

Comparison of function values $f(x) = 2^x$ and $g(x) = (0.5)^x$ :

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

Transformations of Exponential Functions

Exponential functions can be transformed in various ways:

Vertical Translation

The function $f(x) = a \cdot b^x + c$ shifts the graph upward (if $c > 0$ ) or downward (if $c < 0$ ).

Vertical Translation

Comparison of

f(x) = 2^x

with

g(x) = 2^x + 2

Horizontal Translation

The function $f(x) = a \cdot b^{x-h}$ shifts the graph to the right (if $h > 0$ ) or to the left (if $h < 0$ ).

Applications of Exponential Functions

Exponential functions are widely used in daily life:

Population Growth

Living organism populations often follow exponential growth patterns. If the initial population is $P_0$ and the growth rate is $r$ per time period, then:

P(t) = P_0 \cdot (1 + r)^t

Example: Bacterial population that reproduces every hour at a rate of 20%:

Initial population: 1000 bacteria
Growth rate: $r = 0.2$
Function: $P(t) = 1000 \cdot (1.2)^t$

Bacterial Growth Table

Time (hours)	0	1	2	3	4	5
Population	1000	1200	1440	1728	2074	2488

Radioactive Decay

Radioactive substances decay following exponential functions. If the initial mass is $M_0$ and the half-life is $t_{1/2}$ , then:

M(t) = M_0 \cdot \left(\frac{1}{2}\right)^{t/t_{1/2}}

Compound Interest

Investments with compound interest grow exponentially. If the initial capital is $P$ , interest rate is $r$ per year, and time is $t$ years:

A = P \cdot \left(1 + \frac{r}{n}\right)^{nt}

where $n$ is the frequency of interest compounding per year.

Exponential Equations

An exponential equation is an equation that contains a variable in the exponent. General form:

b^x = c

Method 1: Equalizing Bases

If $b^{f(x)} = b^{g(x)}$ , then $f(x) = g(x)$

Example: Solve $2^{x+1} = 8$

2^{x+1} = 8

2^{x+1} = 2^3

x + 1 = 3

x = 2

Method 2: Using Logarithms

To solve $b^x = c$ , use logarithms:

x = \log_b c = \frac{\log c}{\log b}

Exercises

Determine the value of $f(3)$ if $f(x) = 5 \cdot 2^x$
Solve the equation $3^{2x-1} = 27$
A city's population is 50,000 people and grows 3% per year. What will the population be after 10 years?
A radioactive substance has a half-life of 5 years. If the initial mass is 100 grams, how much mass remains after 15 years?

Answer Key

$f(3) = 5 \cdot 2^3 = 5 \cdot 8 = 40$
$3^{2x-1} = 27 = 3^3$ , so $2x - 1 = 3$ , therefore $x = 2$
$P(10) = 50000 \cdot (1.03)^{10} = 50000 \cdot 1.344 = 67.195$ people
$M(15) = 100 \cdot \left(\frac{1}{2}\right)^{15/5} = 100 \cdot \left(\frac{1}{2}\right)^3 = 100 \cdot \frac{1}{8} = 12.5$ grams

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