Problem 10 - College Algebra

Question

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$ f(x)=2 x(x+2)(x-1)^{2} $$

Step-by-Step Solution

Answer

The function \( f(x) = 2x(x+2)(x-1)^{2} \) is a polynomial function.

Step 1: Expand the Function

The provided function is \( f(x) = 2x(x+2)(x-1)^2 \). To identify the type of function, we should first expand this expression.First, expand \((x+2)(x-1)^2\):\[(x-1)^2 = (x-1)(x-1) = x^2 - 2x + 1\]Now, distribute \((x+2)\) across \(x^2 - 2x + 1\):\[x(x^2 - 2x + 1) + 2(x^2 - 2x + 1) = x^3 - 2x^2 + x + 2x^2 - 4x + 2\]This results in:\[x^3 - x + 2\] Finally, multiply this expression by \(2x\):\[2x(x^3 - x + 2) = 2x^4 - 2x^2 + 4x\] Thus, after expansion, we have \( f(x) = 2x^4 - 2x^2 + 4x \).

Step 2: Identify the Function Type

With the expanded expression \( f(x) = 2x^4 - 2x^2 + 4x \), we can identify the type of function.A polynomial function is defined as a function of the form \(a_nx^n + a_{n-1}x^{n-1} + \, ...\, + a_1x + a_0\), where \(n\) is a non-negative integer and the \(a_i\) are constants.The given expression \( 2x^4 - 2x^2 + 4x \) fits this form, with terms \(2x^4, -2x^2,\) and \(4x\). Therefore, \( f(x) \) is a polynomial function.

Key Concepts

Expanding Expressions

When working with functions, expanding expressions helps us understand the function's behavior and classify it correctly. The given function was \( f(x) = 2x(x+2)(x-1)^2 \). To identify the type of function, it was necessary to expand the expression. Expansion involves breaking down complex expressions into simpler components. This simplification can often lead to the discovery of the polynomial structure.

• First, the inner parts are expanded: \((x-1)^2\), which results in \(x^2 - 2x + 1\).
• Next, this expression is distributed through \((x+2)\), giving rise to \(x^3 - 2x^2 + x + 2x^2 - 4x + 2\).
• After combining like terms, the expression simplifies to \(x^3 - x + 2\).
• This is then multiplied by \(2x\), resulting in the expanded form \(2x^4 - 2x^2 + 4x\).

Expanding expressions like this transforms the function into a form where each term is clearly laid out, making it easier to identify the type of function you are dealing with.

Types of Functions

Understanding different types of functions is crucial in mathematics as it helps to determine the function's characteristics and behavior. Functions can be categorized into several types, with polynomial and power functions being among the most common.

• Polynomial Functions: These are expressed in the form \( a_nx^n + a_{n-1}x^{n-1} + \, ...\, + a_1x + a_0 \), where \(n\) is a non-negative integer.
• Power Functions: These have the general form \( f(x) = ax^n \) where \(a\) is a constant and \(n\) is a real number.

Each type of function has unique properties. For example, polynomial functions are smooth and continuous and their graphs are always "nice curves" with no breaks or holes. Recognizing the type of function can aid in understanding its graphical behavior and in solving related algebraic problems.

Function Identification

Function identification involves determining the proper category for a given function. It is essential in solving and analyzing mathematical problems. With the expanded expression \( f(x) = 2x^4 - 2x^2 + 4x \), identification becomes straightforward. Understanding how each function is structured is key to identifying them:

• Since \( f(x) = 2x^4 - 2x^2 + 4x \) can be expressed as a sum of terms each having the form \(a_ix^i\) where the exponents \(i\) are non-negative integers, it fits the polynomial function criteria.
• This function is not a power function because not all terms can be derived from a singular basis of multiplication.

Identifying a function correctly allows mathematicians and students to predict its behaviors, such as growth trends or zero-points, and perform appropriate algebraic manipulations or integrations.