Problem 7 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The function is a power function.

Step 1: Simplify the Given Expression

Let's start by simplifying the expression \(f(x) = (x^2)^3\). Using the exponentiation rule, when raising a power to a power, we multiply the exponents: \(x^{a^b} = x^{a \cdot b}\). Applying this, we get:\[f(x) = x^{2 \times 3} = x^6\].

Step 2: Identify the Type of Function

The simplified expression is \(f(x) = x^6\). A power function is one that can be expressed in the form \(f(x) = kx^a\), where \(k\) is a constant and \(a\) is a real number. In this case, \(k = 1\) and \(a = 6\).Thus, \(f(x) = x^6\) is a power function.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It is in the form:

• \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]

where \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients, \( n \) is a non-negative integer, and \( a_n eq 0 \). This function often combines various terms where each term is defined by the power of \( x \) it contains.
The beauty of polynomial functions lies in their versatility. They can represent simple linear relationships or complex curves. Every polynomial function is a smooth, continuous curve, which makes them important in calculus and real-world modeling applications. They include quadratic, cubic, quartic functions, and more.
In our exercise, after simplifying the expression, we had \( f(x) = x^6 \) which is a polynomial function because it matches the standard form and all coefficients are real numbers.

Exponentiation Rule

The exponentiation rule is a powerful tool in algebra that helps simplify expressions involving exponents. It states that when you raise a power to another power, you multiply the exponents. In mathematical terms, if you have an expression

• \((a^m)^n\)

it simplifies to

• \( a^{m \cdot n} \).

This rule is crucial when working with expressions involving exponents because it reduces more complex power expressions into simpler forms. For example, in the given exercise, using the exponentiation rule on \( f(x) = (x^2)^3 \), we calculated it as \( f(x) = x^{2 \times 3} = x^6 \).
Understanding and applying this rule allows us to manage calculations effectively and convert expressions into forms that are easier to interpret and analyze.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing expressions into their simplest form. This process makes manipulation, analysis, and further calculations easier. When simplifying, we apply different rules to rewrite expressions more efficiently.
In our exercise, the initial expression was \((x^2)^3\). To simplify it, the exponentiation rule was applied, reducing it to \(x^6\). Through simplification:

• We reduce the complexity of expressions.
• It allows easy identification of the function type, like power or polynomial function.
• It helps in solving equations and understanding relationships between variables.

Simplification is especially handy in dealing with any kind of mathematical work, ensuring fewer errors and clearer insight into what the expression represents.