Introduction to rational functions
Textbook sections and practice problems 6.1: 1 – 10 all, 13, 15, 2559 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.

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 t^{2} + 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 2 x^{2} + 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
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 

a^{2}b^{2}=(ab) (a+b) a^{2}+b^{2} 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) = x^{2}
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.