Exponential Functions: Definitions, Properties, and Graphs

Exponential and Polynomial Functions

Definitions and Key Differences

Understanding the distinction between polynomial functions and exponential functions is fundamental in calculus and precalculus. Each type of function has unique properties and applications.

• Polynomial Function: A function where the variable is raised to a constant power. General form: $f(x) = x^n$

• Exponential Function: A function where a constant base is raised to a variable exponent. General form: $f(x) = b^x$

Key Differences:

• In polynomials, the base is the variable and the exponent is a constant.

• In exponentials, the base is a constant and the exponent is the variable.

Examples

• Polynomial: $f(x) = x^2$

• Exponential: $f(x) = 2^x$

Identifying Exponential Functions

Determining Exponential Form

To determine if a function is exponential, check if the variable is in the exponent and the base is a constant.

• Example: $f(x) = (\frac{1}{2})^x$ is exponential. Base: $\frac{1}{2}$ Power: $x$

• Example: $f(x) = 10^{x+1}$ is exponential. Base: $10$ Power: $x+1$

• Example: $f(x) = x^4$ is polynomial, not exponential.

Evaluating Exponential Functions

Substituting Values

To evaluate an exponential function for a given value of $x$, substitute $x$ into the exponent and calculate.

• Example: $f(x) = 2^x$ For $x = 4$: $f(4) = 2^4 = 16$

• For $x = -3$: $f(-3) = 2^{-3} = \frac{1}{8}$

• For $x = 3.14$: $f(3.14) = 2^{3.14}$ (use calculator)

Graphing Exponential Functions

General Properties

Exponential functions have distinctive graphs:

• Continuous and smooth (no breaks or holes)

• Domain: All real numbers ($(-\infty, \infty)$)

• Range: For $b > 0$, $b \neq 1$, the range is $(0, \infty)$

• Horizontal asymptote: $y = 0$

Graph Behavior

• For $b > 1$, the graph increases as $x$ increases.

• For $0 < b < 1$, the graph decreases as $x$ increases.

Table: Comparison of Exponential Graphs

Base $b$	Graph Direction	Steepness
$b > 1$	Increasing	Steeper for larger $b$
$0 < b < 1$	Decreasing	Steeper for smaller $b$

Example Table of Values

$x$	$f(x) = (\frac{1}{2})^x$
-3	8
-2	4
-1	2
0	1
1	0.5
2	0.25
3	0.125

The Number $e$ and Natural Exponential Functions

Definition and Properties

The number $e$ is a mathematical constant approximately equal to 2.71828. It is the base of the natural exponential function and arises in many contexts, including compound interest and calculus.

• Natural Exponential Function: $f(x) = e^x$

• $e$ is not a variable, but a constant.

Evaluating $e^x$

• For $x = 2$: $e^2 \approx 7.389$

• For $x = -3$: $e^{-3} \approx 0.0498$

Graphing $e^x$

• The graph of $f(x) = e^x$ is similar to $f(x) = 2^x$ and $f(x) = 3^x$, but increases more rapidly than $2^x$ and less rapidly than $3^x$.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Horizontal asymptote: $y = 0$

Transformations of Exponential Functions

Graphing Techniques

Transformations allow us to shift, reflect, and stretch exponential graphs.

• Horizontal shift: $f(x) = b^{x-h}$ shifts the graph $h$ units right.

• Vertical shift: $f(x) = b^x + k$ shifts the graph $k$ units up.

• Reflection: $f(x) = -b^x$ reflects the graph over the $x$-axis.

Steps to Graph $g(x) = e^{x-3} - 1$

1. Identify and graph the parent function $f(x) = e^x$.

2. Shift horizontally by $+3$ units (to the right).

3. Shift vertically by $-1$ units (down).

4. Plot key points and sketch the curve approaching the new asymptote $y = -1$.

Summary Table: Exponential Function Properties

Function	Domain	Range	Asymptote
$f(x) = b^x$	$(-\infty, \infty)$	$(0, \infty)$	$y = 0$
$f(x) = b^x + k$	$(-\infty, \infty)$	$(k, \infty)$	$y = k$

Applications of Exponential Functions

Compound Interest and Growth

Exponential functions model real-world phenomena such as population growth, radioactive decay, and compound interest.

• Compound Interest Formula: $A = P e^{rt}$ Where $A$ is the amount, $P$ is the principal, $r$ is the rate, and $t$ is time.

• Population Growth: $N(t) = N_0 e^{kt}$

Additional info: Exponential functions are foundational for understanding calculus topics such as derivatives and integrals of exponential expressions.