Problem 11 - College Algebra

Question

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$ f(x)=3^{x+1} $$

Step-by-Step Solution

Answer

The function is neither a power function nor a polynomial function; it is an exponential function.

Step 1: Define Power and Polynomial Functions

A power function is of the form \( f(x) = kx^n \), where \( k \) and \( n \) are constants. A polynomial function is defined as \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are constants and \( n \) is a non-negative integer.

Step 2: Identify Characteristics of the Given Function

The given function is \( f(x) = 3^{x+1} \). This function involves an exponentiation where the base is a constant, 3, and the exponent is \( x+1 \). This structure does not match the form of a power function or polynomial function because it includes an exponential variable.

Step 3: Conclusion on Function Type

Since the function \( f(x) = 3^{x+1} \) has a variable exponent, it does not match the characteristic forms of either power functions or polynomial functions. It is an exponential function because the base is raised to a power involving a variable.

Key Concepts

Power Functions

Power functions are a fundamental type of mathematical function often encountered in algebra. These functions take the form \( f(x) = kx^n \), where \( k \) is a constant coefficient and \( n \) is a real number exponent.
Power functions have some key features:

• The variable \( x \) is raised to a constant power \( n \).
• The coefficient \( k \) determines the function's growth rate or shape.
• When \( n \) is a whole number, the graph of these functions typically passes through the origin.

Understanding power functions is useful for identifying when variables are multiplied to fixed powers, allowing us to model various real-world scenarios like area and volume calculations.

Polynomial Functions

Polynomial functions represent one of the most common and significant types of functions found in mathematics. They take the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \). Here,

• Each \( a_i \) is a constant coefficient.
• The powers of \( x \) are whole numbers or non-negative integers.
• The degree of the polynomial is the highest power \( n \) of \( x \).

Polynomial functions are pivotal because they can model a broad range of real-world behaviors such as trends and patterns.
Graphs of polynomials can vary widely but usually have smooth, continuous curves without breaks or gaps, making them easily recognizable.

Variable Exponent

A variable exponent means that the power to which the base number is raised involves a variable, such as \( x \) in the expression \( 3^{x+1} \). This concept is fundamental to understanding functions that have an exponential structure rather than polynomial or power structures. In the expression given:

• Unlike power functions, the exponent involves a variable making it not a fixed number.
• This feature describes exponential growth or decay, which is distinct from linear or polynomial growth.

When identifying function types, a variable exponent indicates the function does not conform to the standard forms of power or polynomial functions, often classifying it as an exponential function instead.

Function Identification

Function identification involves determining the type of function based on its equation structure. With the exercise's focus on identifying whether \( f(x) = 3^{x+1} \) is a power or polynomial function, we consider its characteristics:

• A power function requires \( x \) to be raised to a fixed power.
• A polynomial function needs the exponents to be non-negative integers.

In this case, since the exponent contains a variable, \( f(x) = 3^{x+1} \) does not fit any traditional power or polynomial form.
This makes the function an exponential function due to the presence of a variable exponent, illustrating how understanding exponent types is crucial for accurate function classification.