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Introduction to rational functions

Textbook sections and practice problems
6.1: 1 – 10 all, 13, 15, 25-59 odd, 93, 97, 95

Definition
A function f whose formula can be written in the form

where p and q are
both polynomial functions is called a rational function.

Example 1
Graph the function on your calculator and carefully copy the graph onto Figure 1.
State the domain of the function. Finally, write the equation for the function in simplified form.

Recall

To factor x² + x − 12 we first need to find an
integer factor pair of −12 that adds to 1.

Example 2
Graph the function on your calculator and carefully copy the graph onto Figure 2.
State the domain of the function. Finally, write the equation for the function in simplified form.

Recall

To factor t² + 2t − 8 we first need to find an integer factor pair of −8 that adds to 2 .

Example 3
Simplify the formula for Make sure that you state any
necessary domain restrictions. State any other numbers that are not in the domain of g .

Example 4
Simplify . Make sure that you state any necessary domain restrictions.

Recall
To factor x² − 5 x + 4 we first need to find an integer factor pair of 4 that adds to −5

To factor 2 x² + 5 x −12 we first need to find an integer factor pair of (2)(−12) that adds to 5

Example 4
Simplify each rational expression; make sure that you state any necessary restrictions to the
domains.

Simplify

Simplify

Example 5
Simplify . State any necessary restrictions on the domain.

Example 6
Simplify the formula for . What is the domain of f ?


Thinking about what y couldn't be
a²-b²=(a-b) (a+b) a²+b² is psime!

Example 7
Simplify the formula for. State any necessary restrictions on the domain. What
other numbers are not in the domain of g ?

Example 8
Simplify

Example 9

Suppose that f (x) = 3 x + 2 , g (t ) = 2 − 7t , and m(x) = x²

a. Find and simplify f (x + h) , g (t + h) , and m(x + h) .

b. Find and simplify , and

Additional practice problems for you

1. Complete simplify each expression. Make sure that you state all necessary domain restrictions.

2. Simplify each function formula making sure that you state any necessary domain restrictions.
Then state the domain of the function using interval notation.

3. Find and completely simplify f (x + h) for each of the following function.

4. Find and completely simplify for each of the following function.