Download 3.5

Lesson 3.5 – Finding the domain of a
Rational Function
To find the domain set the denominator to zero
and solve for x. The domain will be all real
number except that value for x. x 2  9
The denominator is x- 3.
x 3
Solve for x.
x- 3 = 0
x=3
The domain is all Real Numbers
Except x≠3
Example 2
Find the domain:
x3
x2  9
X2 + 9 = 0
X2 = -9
The domain is only for real numbers not
imaginary the domain is(-∞, ∞)
YOU TRY!!!
Find the domain of
x
x 2  25
x3
Find the domain of 2
x  16
Answers
1. All real numbers except
x≠5 or -5
2. All real numbers.
The parent function of rational functions is
1
f ( x) 
x
What does the graph look like?
Another Basic Rational Function
f(x) = 1/x2 and it looks like this:
Asymptotes: lines that a graph approaches
but does not cross
Vertical asymptotes:
 Whichever values are not allowed in the
domain will be vertical asymptotes on the
graph.
Where is the domain limited?
denominator
Set those factors that only appear in the
denominator or those that appear more
times in the denominator than numerator
equal to zero and solve.
Example
3x  1
1.Find the vertical asymptotes: f ( x) 
x2
Set x – 2 = 0, x = 2 is a vertical asymptote.
x3
2.Find the vertical asymptotes:f ( x)  2
x 9
2
Factor x – 9 = (x-3)(x+3)
x3
1

f ( x) 
x 3
( x  3)( x  3)
Set x – 3 = 0, x =3 is the vertical asymptote
There won’t be one at x=-3, which means there is a
hole in the graph at -3 or point discontinuity.
Picture next slide.
Horizontal asymptotes:
 Look at the degrees of the numerator and
denominator
If the degrees are equal then the horizontal
asymptote is the ratio of the leading coefficients
( y = ratio of leading coefficients)
If the degree in the denominator is greater then
the horizontal asymptote is y = 0
If the degree in the numerator is greater then
there is no horizontal asymptote.
Definition of Horizontal Asymptotes
Ex 1: Find the horizontal asymptotes of
each rational function
 A)
3x  1
f ( x) 
x2
4x
B) f ( x)  2 x 2  1
Horizontal: degrees
Horizontal: If the degree
in the denominator is
greater than the
numerator horizontal
asymptote is y = 0
1st
are equal (both are
degree) so y = ratio 3/1
y=3
C)
4x 3
f ( x)  2
2x  1
If the degree in the numerator is
greater than there is no horizontal
asymptote.
YOU TRY!!! Ex 1: find the vertical &
horizontal asymptotes of each rational
function
 A)
4x  1
f ( x) 
x5
 Vertical: x-5=0
x=5
Horizontal: degrees
are equal (both are 1st
degree) so y = ratio 4/1
y=4
2
x
 3x  10
 B) f ( x) 
( x  1)( x  2)( x  3)
 Vertical:
x+1 = 0
x+3=0
So x = -1, x = -3
 Horizontal: denominator is
bigger (3rd degree vs. 2nd)
so y
=0
x = -2 is a hole
Lesson 3.5 Graphing a Rational
Function
Rational Functions that are not
transformations of f(x) = 1/x or f(x) =
1/x2 can be graphed using the following
suggestions.
Strategy for Graphing a Rational Function
The following strategy can be used to graph f(x) = p(x)
q(x),
Where p and q are polynomial functions with no common
factors.
Seven steps:
Determine whether the graph of
f has symmetry.
f(-x) = f(x): y-axis symmetry
f(-x) = -f(x): origin symmetry
 Step 1:
•Step 2: Find the y-intercept (if there is one) by evaluating
f(0).
•Step 3: Find the x-intercepts (if there are any) by solving
the equation p(x) = 0.
Steps continued:
 Step 4: Find any vertical asymptote(s) by solving
the equation q(x) =0.
•Step 5: Find the horizontal asymptote (if there is one )
using the rule for determining the horizontal asymptote
of a rational function.
•Step 6: Plot at least one point between and beyond
each x-intercept and vertical asymptote.
•Step 7: Use the information obtained previously to
graph the function between and beyond the vertical
asymptotes.
Example 1: Graph:
f(x) = 2x
x-1
Example 2:
2
3x
f ( x)  2
x 4
Step 1: Determine Symmetry: f(-x)
Step 2: Find the Y-intercept. f(0)
Step 3: Find the x-intercepts. p(x) = 0
Step 4: Find the vertical asymptote(s). q(x)=0
Step 5: Find the horizontal asymptote. (Degree of numerator and denominator)
Step 6: Plot points between and beyond each x-intercept and vertical asymptote.
(table)
Step 7: Graph the function.
2
3x
f ( x)  2
x 4
3( x)
3x
1. f ( x) 
 2
2
(  x)  4 x  4
2
2. f(0) = 3*02 =
02 – 4
2
0 = 0
The graph of f is symmetric with
respect to the y-axis.
The y-intercept is 0, so the graph
passes thru the origin.
-4
3. 3x2=0 , so x = 0.
The x-intercepts is 0, verifying
the graph passes through the
origin.
4. Set q(x) =0,
x2-4
= 0, so x = 2 and
x= -2
The vertical asymptotes are x =
-2 and x=2.
Ex. 2 cont:
5. Look at the degree of numerator
and denominator. They are equal
so you use the leading
coefficients. 3/1
The horizontal
asymptote is y = 3.
6. Plot points between and beyond each x-intercept and
vertical asymptote.
X
-3 -1
f(x) 3x2 27/ -1
x2 -4 5
1 3
4
-1 27/5 4
7. Graph the functions.
Summary:
If you are given the equation of a rational
function, explain how to find the vertical
asymptotes of the function.