Download Chapter 11 – Rational Functions

Chapter 11 – Rational Functions
Lesson #2 - Rational Expressions and Functions
Objectives:
-
Introduce the students to rational expressions and functions.
-
Define and illustrate the use of rational expressions and functions.
k
or xy=k. A
x
y
rational expression by definition is a fraction! Rational contains the word Ratio, so y:x or .
x
Below is a good definition of rational expressions.
Review – Lesson #1 used fractions to solve inverse variations of the form y 
Rational Expression
A rational expression is one that
can be written in the form
P
Q
where P and Q are polynomials and Q does not equal 0.
EXPLAINING DOMAIN
With rational functions, we need to watch out for values that cause our denominator to be 0. If
our denominator is 0, then we have an undefined value. So, when looking for the domain of a
given rational function, we use a back door approach. We find the values that we cannot use,
which would be values that make the denominator 0.
Example One: Find all numbers that must be excluded from the domain of
x5
.
x  3x  4
2
Our restriction is that the denominator of a fraction can never be equal to 0.
So to find what values we need to exclude, think of what value(s) of x, if any, would cause the
denominator to be 0.
x 2  3x  4  ( x  1)( x  4)
*Factoring the denominator.
1
Since 1 would make the first factor in the denominator 0, then 1 would have to be excluded.
Since - 4 would make the second factor in the denominator 0, then - 4 would also have to be
excluded.
Example: What is the domain?
A) y 
1
x
B) y 
x2  4
x
x
C) y  2

x2
x  8 x  15 ( x  5)( x  3)
D) y 
x
4
Solutions: A) Domain is anything but 0.
B) Domain is anything but 2.
C) Domain is anything but 5 or 3.
D) Domain is anything! Why?
Trivial Rational Expression: An expression where no variable exists in the denominator. For
x 1
example, in D) above, y   x , or just a regular polynomial.
4 4
“But what if the denominator is zero??”
EXPLAINING ASYMPTOTES
At each value where the denominator is equal to zero, the function is undefined.
Why can you not divide by zero?
Division by zero is an operation for which you cannot find an answer, so it is disallowed.
You can understand why if you think about how division and multiplication are related.
12 divided by 6 is 2 because
6 times 2 is 12
12 divided by 0 is x would mean that
0 times x = 12
But no value would work for x because 0 times any number is 0. So division by
zero doesn't work.
Asymptotes occur when you have a value for which a function is undefined, and where the limit
(the number the function is approaching as it gets close to that value) is either positive or
negative infinity. This could result from a denominator in a function including a variable that
cannot be canceled out with something in the numerator.
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Homework: Pg. 536/#11, 12, 13, 15, 16 (naming undefined)
#18, 21, 24, 27, 30, 33 (evaluating for x=1 and x=-2)
#36, 39, 42 (graphing, list where x is undefined)
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