Problem 9 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The function is neither a power function nor a polynomial function.

Step 1: Analyze the Given Function

The function is given as \( f(x) = \frac{x^2}{x^2 - 1} \). This is a rational function because it is expressed as a ratio of two polynomials.

Step 2: Determine If It's a Power Function

A power function has the form \( f(x) = ax^b \), where \( a \) and \( b \) are constants. Since \( f(x) = \frac{x^2}{x^2 - 1} \) does not strictly match this format due to the presence of the denominator, it is not a power function.

Step 3: Determine If It's a Polynomial Function

A polynomial function consists of terms of the form \( a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0 \). Since the function \( f(x) = \frac{x^2}{x^2 - 1} \) involves division by a polynomial \( x^2 - 1 \), it cannot be simplified to a normal polynomial form and hence is not a polynomial function.

Step 4: Conclude the Classification

Since the function does not meet the criteria for either a power function or a polynomial function, it is classified as neither based on its current form as a ratio of two polynomials.

Key Concepts

Power Functions

Power functions are simple yet important mathematical expressions. They have the general form \( f(x) = ax^b \), where:

• \( a \) is a nonzero constant.
• \( b \) can be any real or complex number.

The key characteristic of power functions is that they don't involve any additions or subtractions of other terms beyond \( ax^b \). This makes them very straightforward, involving a single term.
For example, \( f(x) = 3x^2 \) is a power function because it follows the form of \( ax^b \), with \( a = 3 \) and \( b = 2 \).
However, if there is any kind of addition or subtraction involved, or variables appear in the denominator, the function cannot be classified as a power function. That's why our given function, \( f(x) = \frac{x^2}{x^2 - 1} \), is not a power function.

Polynomial Functions

Polynomial functions are a cornerstone in algebra and calculus. They are sums of multiple terms, each consisting of a coefficient multiplied by a variable raised to a non-negative integer power. The structure of a polynomial function is:

• \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)
• Where \( a_i \) are coefficients, and \( n \) is the degree of the polynomial.

Polynomial functions can appear very simple, such as \( f(x) = 2x + 3 \), or be quite complex with many terms. The key is they include terms only involving non-negative powers of \( x \) and do not have variables in the denominator.
In the given function \( f(x) = \frac{x^2}{x^2 - 1} \), the denominator \( x^2 - 1 \) prevents it from being a polynomial. Because polynomial terms cannot be divided by other polynomial expressions without altering this fundamental structure.
Thus, this function doesn't fit the criteria for polynomial functions.

Function Classification

Classifying functions helps us understand their behavior and the operations we can perform on them. The three primary classifications for functions are:

• Power Functions: Specific single-term forms \( ax^b \).
• Polynomial Functions: Sum of terms with non-negative integer powers of \( x \).
• Rational Functions: Ratios of two polynomials.

Once we identify the form of a function, we can study its properties like continuity, differentiability, and integrability more effectively.
For \( f(x) = \frac{x^2}{x^2 - 1} \), it doesn't neatly fit into the first two categories—power or polynomial—due precisely to the division by a polynomial. Therefore, it is classified as a rational function.
Rational functions are versatile but can have unique behaviors, like vertical asymptotes, arising from divisions by zero in the denominator.