Polynomial and Exponential Functions: Definitions, Properties, and Examples

Polynomial and Exponential Functions

Introduction

Understanding the distinction between polynomial functions and exponential functions is fundamental in College Algebra. These two types of functions differ primarily in the placement of the variable: in polynomials, the variable is the base, while in exponentials, the variable is the exponent.

Polynomial Functions

Polynomial functions are algebraic expressions where the variable appears in the base and is raised to a constant, positive integer exponent.

• General Form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$

• Example: $f(x) = x^2$

• Key Properties:

  • The base is a variable (e.g., $x$).

  • The exponent is a constant integer (e.g., $2$).

Exponential Functions

Exponential functions are expressions where the variable appears in the exponent, and the base is a constant.

• General Form: $f(x) = a \cdot b^x$ where $a \neq 0$, $b > 0$, and $b \neq 1$

• Example: $f(x) = 2^x$

• Key Properties:

  • The base is a constant (e.g., $2$).

  • The exponent contains the variable (e.g., $x$).

Function Type	Base	Exponent	Variable Location
Polynomial	Variable	Constant	Base
Exponential	Constant	Variable	Exponent

Identifying Exponential Functions

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

• Example 1: $f(x) = (\frac{1}{3})^x$ is exponential (base: $\frac{1}{3}$, exponent: $x$).

• Example 2: $f(x) = 17^0$ is not exponential (exponent is constant).

• Example 3: $f(x) = 10^{x+1}$ is exponential (base: $10$, exponent:$x+1$).

Evaluating Exponential Functions

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

• Example: Evaluate $f(x) = 2^x$ for $x = 4$:

$f(4) = 2^4 = 16$

• Example: Evaluate $f(x) = 2^x$ for $x = -3$:

$f(-3) = 2^{-3} = \frac{1}{8}$

• Example: Evaluate $f(x) = 2^x$ for $x = 3.14$:

$f(3.14) \approx 8.815$

• Example: Evaluate $f(x) = 2^x$ for $x = 12$:

$f(12) = 4096$

Practice: Identifying and Evaluating Exponential Functions

Function	Exponential?	Base	Exponent	f(4)
$f(x) = (-2)^x$	Yes	-2	x	$(-2)^4 = 16$
$f(x) = 3(1.5)^x$	Yes	1.5	x	$3(1.5)^4 = 3(5.0625) = 15.1875$
$f(x) = (\frac{1}{4})^x$	Yes	$\frac{1}{4}$	x	$(\frac{1}{4})^4 = \frac{1}{256}$

Summary

• Polynomial functions have a variable base and constant exponent.

• Exponential functions have a constant base and variable exponent.

• To evaluate exponential functions, substitute the given value for the variable in the exponent and compute the result.