Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$ f(x)=x^{5} $$

Step-by-Step Solution

Answer

The function \( f(x) = x^5 \) is both a power function and a polynomial function.

Step 1: Understand the Function Given

We are given the function \( f(x) = x^5 \). Our task is to identify whether this function is a power function, a polynomial function, or neither.

Step 2: Define a Power Function

A power function is of the form \( f(x) = kx^n \), where \( k \) is a constant and \( n \) is a real number. The function \( f(x) = x^5 \) has the form \( f(x) = kx^n \) with \( k = 1 \) and \( n = 5 \) (both are constants), so it satisfies the form of a power function.

Step 3: Define a Polynomial Function

A polynomial function is any function of the form \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0 \), where each \( a_i \) is a constant, and \( n \) is a non-negative integer. In our function \( f(x) = x^5 \), \( a_5 = 1 \) and all other coefficients are 0.

Step 4: Classify the Function

Since the function \( f(x) = x^5 \) can be expressed in the form of both a power function and a polynomial function \( a_5 x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 0 \), it qualifies as both. A power function is a specific type of polynomial function, hence \( f(x) = x^5 \) is a polynomial function. However, since it also meets the criteria for a power function, it is accurately classified as a power function too.

Key Concepts

Power Function

A power function is a simple type of function where the variable is raised to a constant power. More formally, it has the structure \( f(x) = kx^n \). Here, \( k \) is a constant, and \( n \) is any real number.

Understanding power functions is essential because they represent a basic form of mathematical operations found in various real-world applications. Let's break this down further:

• The base of the power function is always the variable \( x \).
• The exponent \( n \) determines the degree of the power function.
• If \( k = 1 \), the function simplifies to just \( x^n \), as seen in the example \( f(x) = x^5 \), making it straightforward.
• Power functions are highly versatile. By changing \( n \), they can quickly represent different relationships, such as quadratic or cubic patterns.

Function Identification

In mathematics, identifying the type of function you are working with is crucial for understanding its properties and behavior.

When given a function like \( f(x) = x^5 \), determining if it's a power function, a polynomial function, or neither helps in understanding how it behaves. Here's how to go about it:

• Check if the function fits the definition of a power function \( f(x) = kx^n \). If it does, then it can be classified as a power function.
• Next, see if it fits the definition for a polynomial function. This involves checking if it can be expressed as a sum of terms like \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0 \).
• If a given term matches both these forms, like in \( f(x) = x^5 \), then it belongs to both categories.

Understanding these classifications not only helps in solving problems more efficiently but also provides a deeper insight into the function's structure and potential transformations.

Mathematical Definitions

Grasping the strict definitions of various mathematical functions provides a strong foundation in learning higher-level mathematics. Let's explore the definitions relevant to our problem:

• Power Function: It's defined as \( f(x) = kx^n \) where the variable \( x \) is raised to a fixed power \( n \). Each function component, like \( k \) (the coefficient) and \( n \) (the exponent), helps describe the function's characteristics.
• Polynomial Function: This is a broader classification which encompasses functions like power functions. A general polynomial function looks like \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \). It's important to note that each coefficient \( a_i \) needs to be a constant, and \( n \) should be a non-negative integer.
• Both types of functions can have special characteristics based on their coefficients and exponents, affecting their graphs and real-world applications.

By mastering these definitions, students can accurately categorize and manipulate functions, paving the way for more complex mathematical explorations.