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Rational Functions

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Rational Functions

Macon State College

Gaston Brouwer, Ph.D.

June 2010

Georgia Performance Standards

Mathematics 4

MM4A1. Students will explore rational functions.

- Investigate and explain characteristics of rational
- functions, including domain, range, zeros, points of
- discontinuity, intervals of increase and decrease, rates
- of change, local and absolute extrema, symmetry,
- asymptotes, and end behavior.

b. Find inverses of rational functions, discussing domain

and range, symmetry, and function composition.

c. Solve rational equations and inequalities analytically,

graphically, and by using the appropriate technology.

- Basics
- What is a Rational Function?
- Domain
- Horizontal & Vertical Asymptotes
- Zeros
- Graphing a Rational Function
- Solving Rational Equations
- Inverses
- Range
- Solving Rational Inequalities

Multiplying fractions:

Adding fractions:

Simplifying fractions:

Definition

A rational function can be written in the form

Where and are both polynomial

functions and

Rational function

Rational function

Not a rational function

The domain of a rational function

is given by:

Examples

Domain:

Domain:

Let be a rational function. The line

is a horizontal asymptote (HA) if:

1. Divide and by the highest power of that

shows up in . Call the resulting functions and

.

2. HA:

HA:

HA:

No Horizontal Asymptote

- If degree( ) < degree( ), the HA is given by

2. If degree( ) = degree( ), the HA is given by

3. If degree( ) > degree( ), there is no HA.

Degree( ) = degree( )=2, so:

HA:

Degree( ) > degree( ), so there is no HA.

Let be a rational function and let

Then the end behavior of is the same

as the end behavior of:

Consider the function

Degree( ) > degree( ), so there is no HA.

Its end behavior is the same as

Let be a rational function.

The line is a vertical asymptote (VA) if:

1. Reduce the function to lowest terms.

2. The vertical asymptote(s) is (are):

where is (are) the solution(s) to

Solve:

VA:

(Note that is not a vertical asymptote!)

1. Reduce the function to lowest terms.

2. The zeros of the rational function are the

solutions to

Find the zeros of

1. Reduce the function to lowest terms.

2. Set the numerator equal to zero and solve

Graph:

1. Reduce to lowest terms:

2. Find y-intercepts (set x=0):

3. Find zeros/x-intercepts (solve f(x)=0):

4. Find the horizontal asymptote:

5. Find the vertical asymptote(s):

6. Create a table for

Not in the domain! (open circle)

Solve

Multiply both sides by

On the TI83/84 calculator:

Solve

Multiply both sides by

No solution

Find the inverse of

1. Write the function in the form y=…

2. Interchange x and y

3. Solve for y

3. Write in the form

1. Read from graph, or

2. Use the fact that:

Find the range of

Previously we found that

Domain of :

Range of :

Solve

Write the equation in the form:

On a number line, mark all the points where

with a “0” and all the points where with a “?”.

Then determine the sign of

on each interval

by using test points.

On the TI83/84: