Rational Functions: Examples, Factoring, Graphing, and Operations

I. What is a Rational Function?

Definition: A rational function is a function that can be expressed as the ratio of two polynomials, \(f(x)=\frac{P(x)}{Q(x)}​\), where \(P(x)\) and \(Q(x)\) are polynomials. The denominator polynomial \(Q(x)\) cannot be the zero polynomial. A basic example is \(f(x) = \frac{1}{x}\)

II. Basic Example of a Rational Function:

Consider the rational function: \(f(x)=\frac{2x+3​}{x^2−4}\)

In this example, \(P(x)=2x+3\) and \(Q(x)=x^2−4\).

III. Factoring Rational Functions:

Purpose: Factoring rational functions is crucial for simplification and understanding the behavior of the function. Follow these steps:

1. Factor Numerator and Denominator:

   • Factor \(P(x)\) and \(Q(x)\) completely.

2. Cancel Common Factors:

   • Cancel common factors between the numerator and denominator.

\(f(x)=\frac{2x+4​}{x^2−4}\)

1. Factor numerator and denominator: \(f(x)=\frac{2x+3}{(x−2)(x+2)}\)​
2. Cancel common factor: \(f(x)=\frac{2\color{red}{(x+2)}}{(x−2)\color{red}{(x+2)}}=\frac{2}{x-2}\)

3. Holes: Any factor canceled introduces a hole in the graph, or a restriction to the domain of the function. In this case, we have a hole where \(x+2=0\), or at \(x=-2\).

IV. Graphing Rational Functions:

Characteristics:

1. Vertical Asymptotes:

   • Set the denominator equal to zero and solve for \(x\). Vertical asymptotes occur where the denominator is zero.

2. Horizontal Asymptotes:

   • Compare the degrees of the numerator and denominator.
     • If degree of \(P(x)\) < degree of \(Q(x)\), the x-axis (y = 0) is the horizontal asymptote.
     • If degree of \(P(x)\) = degree of \(Q(x)\), divide the leading coefficients.
     • If degree of \(P(x)\) > degree of \(Q(x)\), there is no horizontal asymptote.

3. Holes:

   • Factor and cancel common factors. Any factor canceled introduces a hole in the graph.

Example: \(f(x)=\frac{2x+3}{x^2−4}​\)

1. Vertical asymptotes: \(x=2\) and \(x=−2\) (where the denominator equals zero).
2. Horizontal asymptotes: \(y=0\) (degree of numerator < degree of denominator).
3. Hole: None (no common factors canceled).

V. Multiplying and Dividing Rational Functions:

Multiplication: \(\frac{A(x)}{B(x)}\cdot\frac{C(x)}{D(x)}=\frac{A(x)C(x)}{B(x)D(x)}\)

Division: \(\frac{A(x)}{B(x)}\div\frac{C(x)}{D(x)}=\frac{A(x)}{B(x)}\cdot\frac{D(x)}{C(x)}\)

Example: \(\frac{2x+3}{x^2-4}\cdot\frac{x+1}{x-2}=\frac{(2x+3)(x+1)}{(x-2)(x+2)}\)

VI. Addition and Subtraction of Rational Functions:

Addition: \(\frac{A(x)}{B(x)}+\frac{C(x)}{D(x)}=\frac{A(x)D(x)+C(x)B(x)}{B(x)D(x)}\)

Subtraction: \(\frac{A(x)}{B(x)}-\frac{C(x)}{D(x)}=\frac{A(x)D(x)-C(x)B(x)}{B(x)D(x)}\)

Example: \(f(x)=\frac{2}{x+1}+\frac{3x}{x^2-1}\)

Steps:

1. Find a Common Denominator:

   The common denominator is the product of the individual denominators. In this case, the denominators are \(x+1\) and \(x^2−1\). The factored form of \(x^2−1\) is \((x+1)(x−1)\), so the common denominator is \((x+1)(x−1)\).

2. Adjust the Numerators:

   Adjust each fraction's numerator to have the common denominator:

   • For \(\frac{2}{x+1}\)​, multiply the numerator and denominator by \((x−1)\) to get \(\frac{2(x-1)}{(x+1)(x-1)}\)​.
   • For \(\frac{3x}{x^2-1}\)​, no adjustment is needed as the denominator is already \((x+1)(x−1)\).

3. Add the Numerators:

   \(\frac{2(x-1)}{(x+1)(x-1)}+\frac{3x}{(x+1)(x-1)}\)

   Combine the numerators over the common denominator: 
   \(\frac{2(x-1)+3x}{(x+1)(x-1)}\)

4. Simplify:

   Simplify the numerator: \(\frac{2x-2+3x}{(x+1)(x-1)}\)​

   Combine like terms in the numerator: \(\frac{5x-2}{(x+1)(x-1)}\)

So, the sum of \(\frac{2}{x+1}+\frac{3x}{x^2-1}\) is \(\frac{5x-2}{(x+1)(x-1)}\)​.

Summary:

Rational functions are expressed as the ratio of two polynomials, and understanding them involves factoring, graphing, and performing operations like multiplication and division. Identifying asymptotes, holes, and simplifying expressions are key steps in working with rational functions. The characteristics of rational functions offer insights into their behavior and help create accurate graphs.