2. Objectives:
οfind the domain, range, zeroes
and intercepts of rational
functions
οdetermine the vertical and
horizontal asymptotes of rational
function.
3. ο The domain of a rational function π π₯ =
π(π₯)
π·(π₯)
is all the values of that π₯ will not make π·(π₯)
equal to zero.
ο To find the range of rational function is by
finding the domain of the inverse function.
ο Another way to find the range of rational
function is to find the value of horizontal
asymptote.
Domain and Range of Rational Function
6. π π =
π
π β π
π β π = π
Focus on the
denominator
The domain of π(π)
is the set of all real
numbers except π.
EXAMPLE 1:
π = π
To find the domain:
π«: π π β β, π β π
7. π π =
π
π β π
οΌ Change π(π₯) into y
EXAMPLE 1:
To find the range:
π =
π
π β π
οΌ Interchange the position
of x and y
π =
π
π β π
οΌ Simplify the rational
expression
π π β π = π
ππ β ππ = π
οΌ Solve for y in terms of x ππ = π + ππ
ππ
π
=
π + ππ
π
π =
π + ππ
π
οΌ Equate the
denominator
to 0.
π = π
The range of π(π) is the set
of all real numbers except π.
πΉ: π π β β, π β π
9. π π =
π β π
π + π
π + π = π
Focus on the
denominator
The domain of π(π)
is the set of all real
numbers except βπ.
EXAMPLE 2:
π = βπ
To find the domain:
π«: π π β β, π β βπ
10. π π =
π β π
π + π
οΌ Change π(π₯) into y
EXAMPLE 2:
To find the range:
π =
π β π
π + π
οΌ Interchange the position
of x and y
π =
π β π
π + π
οΌ Simplify the rational
expression
π π + π = π β π
ππ + ππ = π β π
οΌ Solve for y in terms of x ππ β π = βπ β ππ
π(π β π)
π β π
=
βπ β ππ
π β π
π =
βπ β ππ
π β π
οΌ Equate the
denominator
to 0.
π β π = π
The range of π(π) is
the set of all real
numbers except π.
πΉ: π π β β, π β π
π(π β π) = βπ β ππ
π = π
12. π π =
π + π
ππ β π
ππ β π = π
Focus on the
denominator
The domain of π(π)
is the set of all real
numbers except π.
EXAMPLE 3:
ππ = π
To find the domain:
π«: π π β β, π β π
π = π
13. π π =
π + π
ππ β π
οΌ Change π(π₯) into y
EXAMPLE 3:
To find the range:
π =
π + π
ππ β π
οΌ Interchange the position
of x and y
π =
π + π
ππ β π
οΌ Simplify the rational
expression
π ππ β π = π + π
2ππ β ππ = π + π
οΌ Solve for y in terms of x πππ β π = π + ππ
π(ππ β π)
ππ β π
=
π + ππ
ππ β π
π =
βπ + ππ
ππ β π
οΌ Equate the
denominator
to 0.
ππ β π = π
The range of π(π) is the set of all
real numbers except
π
π
.
πΉ: π π β β, π β
π
π
π ππ β π = π + ππ
ππ = π
π =
π
π
15. They are the restrictions on the x
β values of a reduced rational
function. To find the restrictions,
equate the denominator to 0 and
solve for x.
Finding the Vertical Asymptotes
of Rational Functions
16. Let n be the degree of the numerator and m
be the degree of denominator:
β’ If π < π, π = π.
β’ If π = π, π =
π
π
, where π is the leading
coefficient of the numerator and π is the
leading coefficient of the denominator.
β’ If π > π , there is no horizontal
asymptote.
Finding the Horizontal Asymptotes
of Rational Functions
17. Find the Degree of Polynomial.
ππ π
π«πππππ
π β π π
πππ
β π β π π
21. π π =
π
π β π
To find the vertical
asymptote:
π β π = π
π = π
Focus on the
denominator
The vertical asymptote
is π = π.
EXAMPLE 1:
22. π π =
π
π β π
To find the horizontal
asymptote:
π < π
Focus on the degree
of the numerator
and denominator
The horizontal asymptote
is π = π.
0
1
EXAMPLE 1:
24. π π =
ππ β π
π + π
To find the vertical
asymptote:
π + π = π
π = βπ
Focus on the
denominator
The vertical asymptote
is π = βπ.
EXAMPLE 2:
25. π π =
ππ β π
π + π
To find the horizontal
asymptote:
π = π
Focus on the degree
of the numerator
and denominator
The horizontal
asymptote is π = π.
1
1
EXAMPLE 2:
π =
π
π
=
π
π
= π
a is the leading coefficient of 4x
b is the leading coefficient of x
27. π π =
ππ + π
πππ + ππ + π
To find the vertical asymptote:
πππ + ππ + π = π
Focus on the
denominator
The vertical
asymptote are π = β
π
π
and π = βπ.
EXAMPLE 3:
ππ + π π + π = π
ππ + π = π π + π = π
ππ = βπ
π = β
π
π
π = βπ
28. 1
2
EXAMPLE 3:
π π =
ππ + π
πππ + ππ + π
To find the horizontal
asymptote:
π < π
Focus on the degree
of the numerator
and denominator
The horizontal asymptote
is π = π.
30. π π =
πππ
β π
ππ + ππ β π
To find the vertical asymptote:
ππ + ππ β π = π
Focus on the
denominator
The vertical
asymptote are π = βπ
and π = π.
EXAMPLE 4:
π + π π β π = π
π + π = π π β π = π
π = βπ π = π
π = βπ
31. 3
2
EXAMPLE 4:
To find the horizontal
asymptote:
π > π
Focus on the degree
of the numerator
and denominator
The rational function has
no horizontal asymptote.
π π =
πππ
β π
ππ + ππ β π
33. Finding the Zeros of Rational
Functions
Steps:
1. Factor the numerator and denominator.
2. Identify the restrictions.
3. Identify the values of x that make the
numerator equal to zero.
4. Identify the zero of f(x).
42. ο Intercepts are x and y β coordinates of
the points at which a graph crosses the
x-axis or y-axis, respectively.
ο y-intercept is the y-coordinate of the
point where the graph crosses the y-
axis.
ο x-intercept is the x-coordinate of the
point where the graph crosses the x-
axis.
Note: Not all rational functions have both x and y intercepts. If the
rational function has no real solution, then it does not have intercepts.
43. Rule to find the Intercepts
1) To find the y-intercept, substitute 0
for x and solve for y or f(x).
2) To find the x-intercept, substitute 0
for y and solve for x.