How to Analyze and Understand Rational Functions: A Comprehensive Guide

Step-by-step Guide to Analyze and Understand Rational Functions

A rational function is a function of the form \(\frac{f(x)}{g(x)}\)​ where \(f(x)\) and \(g(x)\) are polynomials, and \(g(x)≠0\).

Here’s a comprehensive step-by-step guide to understanding these functions:

Step 1: Know the Basics

• Polynomial: A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Example: \(f(x)=2x^3−4x^2+x−7\).
• Rational Function: It’s a ratio of two polynomials. Example: \(R(x)=\frac{x^2-1}{x^2+x-6}\)​.

Step 2: Simplify the Function

Often, you can simplify a rational function by factoring the numerator and the denominator and then canceling out common factors.

Example: \(R(x)=\frac{x^2-1}{x^2+x-6}\)​

Factoring: \(R(x)=\frac{(x−1)(x+1)​}{(x+3)(x−2)}\)

Step 3: Identify the Domain

The domain of a rational function is all real numbers except where the denominator is zero.

To find these exclusions:

1. Set the denominator equal to zero.
2. Solve for \(x\).

Using the previous example: \(x^2+x−6=0\)

The solutions are \(x=2\) and \(x=−3\).

So, the domain of \(R(x)\) is all real numbers except \(x=2\) and \(x=−3\).

Step 4: Find Horizontal Asymptotes

A horizontal asymptote describes the behavior of a rational function as \(x\) approaches positive or negative infinity.

For rational functions:

1. If the degree of the numerator is less than the degree of the denominator, \(y = 0\) is the horizontal asymptote.
2. If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.
3. If the degree of the numerator is greater by one than the denominator, there’s no horizontal asymptote, but there might be an oblique or slant asymptote.

For our example, the degrees of the numerator and the denominator are the same, so the horizontal asymptote is \(y=\frac{1}{1}​=1\).

Step 5: Find Vertical Asymptotes

Vertical asymptotes occur where the function is undefined – precisely where the denominator is zero, but the numerator isn’t.

From Step \(3\), we found that the denominator is zero at \(x=2\) and \(x=−3\). Neither of these makes the numerator zero, so \(x=2\) and \(x=−3\) are vertical asymptotes.

Step 6: Identify Holes (Point Discontinuities)

If a value is excluded from the domain because it makes both the numerator and the denominator zero (after simplification), then that point is a hole in the graph.

Using our example, since there aren’t any values that make both the simplified numerator and denominator zero, there aren’t any holes.

Step 7: Analyze End Behavior and Intercept

• End Behavior: Observe how the function behaves as \(x\) approaches positive and negative infinity by looking at the horizontal or slant asymptotes.
• \(x\)-intercepts: Points where \(y=0\). Set the numerator equal to zero and solve for \(x\).
• \(y\)-intercepts: Points where \(x=0\). Plug \(x=0\) into the function.

Step 8: Sketch the Graph

1. Plot any intercepts.
2. Plot asymptotes as dashed lines.
3. Observe the end behavior.
4. Check a few points between vertical asymptotes to determine the shape of the curve.
5. Draw the curve ensuring it approaches the asymptotes and passes through any intercepts.

Using software or a graphing calculator can help verify the sketch.

Step 9: Interpret the Function

• Where is the function positive or negative?
• Is the function increasing or decreasing in certain intervals?
• Understand the real-world implications if the function models a real-world scenario.

Remember that each rational function may have its unique characteristics, so it’s always crucial to analyze each one individually. As you gain experience, you’ll become more comfortable with spotting features and trends in these functions.

Examples:

Example 1:

Given: \(R(x)=\frac{2x+1​}{x^2−1}\). Find the domain and vertical asymptotes.

Solution:

To find the domain, set the denominator \(x^2−1\) not equal to zero: \(x^2−1≠0\)

Factoring: \((x−1)(x+1)≠0\)

This gives \(x≠1\) and \(x≠−1\).

Domain: All real numbers except \(x=1\) and \(x=−1\).

The vertical asymptotes are the x-values excluded from the domain:

Vertical Asymptotes: \(x=1\) and \(x=−1\).

Example 2:

Given: \(R(x)=\frac{x^2+4x+3​}{x^2+x−6}\). Find the domain and vertical asymptotes.

Solution:

To find the domain, set the denominator \(x^2+x−6\) not equal to zero: \(x^2+x−6≠0\)

Factoring: \((x+3)(x−2)≠0\)

This gives \(x≠−3\) and \(x≠2\).

Domain: All real numbers except \(x≠−3\) and \(x≠2\).

The vertical asymptotes are the x-values excluded from the domain:

Vertical Asymptotes: \(x≠−3\) and \(x≠2\).