Problem 7 - College Algebra

Question

Identify the function as a power function, a polynomial function, or neither. $$f(x)=\left(x^{2}\right)^{3}$$

Step-by-Step Solution

Answer

It is a power function and a polynomial function.

Step 1: Simplify the Expression

First, simplify the expression \( f(x) = (x^2)^3 \). Use the power of a power property, which states that \((a^m)^n = a^{m imes n}\). Here, \( a = x \), \( m = 2 \), and \( n = 3 \). So, the expression simplifies to \( f(x) = x^{2 \times 3} = x^6 \).

Step 2: Determine the Type of Function

Now that you have the simplified form \( f(x) = x^6 \), determine the type of function. A power function is of the form \( f(x) = ax^n \) where \( a eq 0 \) and \( n \) is a real number. A polynomial function is a finite sum of the form \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \). The function \( f(x) = x^6 \) fits both definitions with \( a = 1 \) and \( n = 6 \).

Key Concepts

Power function

A power function is a very specific type of mathematical relationship where the function is expressed in the form \( f(x) = ax^n \). Here, \( a \) represents a constant coefficient and \( n \) is an exponent that can be any real number. The key characteristics of power functions are:

• The base \( x \) is raised to a power \( n \).
• \( a \) is not zero.

For the function \( f(x) = x^6 \), every term fits perfectly in this format with \( a = 1 \) and \( n = 6 \).
This means the base \( x \) is multiplied by itself 6 times. Power functions can describe various phenomena, like area (as in \( x^2 \)) or volume (as in \( x^3 \)).
Power functions are a subset of polynomial functions when \( n \) is a positive integer.

Exponentiation

Exponentiation is the mathematical operation involving a base and an exponent. It represents repeated multiplication in a concise form. For instance, \((x^2)^3\) indicates multiplying \(x^2\) three times by itself.

Using the rule of powers, one simplifies this to \( x^{2 \times 3} = x^6 \). The number \( 6 \) here is not a simple number; it conveys that \( x \) is to be multiplied six times.

When working with exponentiation, there are key properties to keep in mind:

• Multiplication of powers: \( a^m \times a^n = a^{m+n} \)
• Power of a power: \((a^m)^n = a^{m \times n} \)

These rules simplify many algebraic expressions and make it easier to work with equations involving powers.

Function identification

In mathematics, identifying a function's type is crucial as it determines how the function behaves and influences how it can be manipulated and calculated.

For the function \( f(x) = x^6 \), we need to consider whether it fits the definition of a power function or a polynomial function. Here's how you can identify it:

• Power function: Takes the form \( ax^n \) with \( a eq 0 \) and \( n \) a real number. \( f(x) = x^6 \) is indeed a power function with \( a = 1 \) and \( n = 6 \).
• Polynomial function: A sum of terms, each involving powers of \( x \) with non-negative integer exponents. \( f(x) = x^6 \) fits this definition as well, as it can be perceived as a single-term polynomial.

By accomplishing these checks, one can confidently determine that \( f(x) = x^6 \) is both a power and a polynomial function. Understanding these categorizations helps in further using derivatives, integrals, and solving equations.