Domain, Range, Zeros,
Intercepts and
Asymptotes of
Rational Function

Objectives:
find the domain, range, zeroes
and intercepts of rational
functions
determine the vertical and
horizontal asymptotes of rational
function.

 The domain of a rational function 𝑓 𝑥 =
𝑁(𝑥)
𝐷(𝑥)
is all the values of that 𝑥 will not make 𝐷(𝑥)
equal to zero.
 To find the range of rational function is by
finding the domain of the inverse function.
 Another way to find the range of rational
function is to find the value of horizontal
asymptote.
Domain and Range of Rational Function

𝒇 𝒙 =
𝟐
𝒙 − 𝟑
EXAMPLE 1:

𝒇 𝒙 =
𝟐
𝒙 − 𝟑
𝒙 − 𝟑 = 𝟎
Focus on the
denominator
The domain of 𝒇(𝒙)
is the set of all real
numbers except 𝟑.
EXAMPLE 1:
𝒙 = 𝟑
To find the domain:
𝑫: 𝒙 𝒙 ∈ ℝ, 𝒙 ≠ 𝟑

𝒇 𝒙 =
𝟐
𝒙 − 𝟑
 Change 𝑓(𝑥) into y
EXAMPLE 1:
To find the range:
𝒚 =
𝟐
𝒙 − 𝟑
 Interchange the position
of x and y
𝒙 =
𝟐
𝒚 − 𝟑
 Simplify the rational
expression
𝒙 𝒚 − 𝟑 = 𝟐
𝒙𝒚 − 𝟑𝒙 = 𝟐
 Solve for y in terms of x 𝒙𝒚 = 𝟐 + 𝟑𝒙
𝒙𝒚
𝒙
=
𝟐 + 𝟑𝒙
𝒙
𝒚 =
𝟐 + 𝟑𝒙
𝒙
 Equate the
denominator
to 0.
𝒙 = 𝟎
The range of 𝒇(𝒙) is the set
of all real numbers except 𝟎.
𝑹: 𝒚 𝒚 ∈ ℝ, 𝒚 ≠ 𝟎

𝒇 𝒙 =
𝒙 − 𝟓
𝒙 + 𝟐
EXAMPLE 2:

𝒇 𝒙 =
𝒙 − 𝟓
𝒙 + 𝟐
𝒙 + 𝟐 = 𝟎
Focus on the
denominator
The domain of 𝒇(𝒙)
is the set of all real
numbers except −𝟐.
EXAMPLE 2:
𝒙 = −𝟐
To find the domain:
𝑫: 𝒙 𝒙 ∈ ℝ, 𝒙 ≠ −𝟐

𝒇 𝒙 =
𝒙 − 𝟓
𝒙 + 𝟐
 Change 𝑓(𝑥) into y
EXAMPLE 2:
To find the range:
𝒚 =
𝒙 − 𝟓
𝒙 + 𝟐
 Interchange the position
of x and y
𝒙 =
𝒚 − 𝟓
𝒚 + 𝟐
 Simplify the rational
expression
𝒙 𝒚 + 𝟐 = 𝒚 − 𝟓
𝒙𝒚 + 𝟐𝒙 = 𝒚 − 𝟓
 Solve for y in terms of x 𝒙𝒚 − 𝒚 = −𝟓 − 𝟐𝒙
𝒚(𝒙 − 𝟏)
𝒙 − 𝟏
=
−𝟓 − 𝟐𝒙
𝒙 − 𝟏
𝒚 =
−𝟓 − 𝟐𝒙
𝒙 − 𝟏
 Equate the
denominator
to 0.
𝒙 − 𝟏 = 𝟎
The range of 𝒇(𝒙) is
the set of all real
numbers except 𝟏.
𝑹: 𝒚 𝒚 ∈ ℝ, 𝒚 ≠ 𝟏
𝒚(𝒙 − 𝟏) = −𝟓 − 𝟐𝒙
𝒙 = 𝟏

𝒇 𝒙 =
𝟕 + 𝒙
𝟐𝒙 − 𝟔
EXAMPLE 3:

𝒇 𝒙 =
𝟕 + 𝒙
𝟐𝒙 − 𝟔
𝟐𝒙 − 𝟔 = 𝟎
Focus on the
denominator
The domain of 𝒇(𝒙)
is the set of all real
numbers except 𝟑.
EXAMPLE 3:
𝟐𝒙 = 𝟔
To find the domain:
𝑫: 𝒙 𝒙 ∈ ℝ, 𝒙 ≠ 𝟑
𝒙 = 𝟑

𝒇 𝒙 =
𝟕 + 𝒙
𝟐𝒙 − 𝟔
 Change 𝑓(𝑥) into y
EXAMPLE 3:
To find the range:
𝒚 =
𝟕 + 𝒙
𝟐𝒙 − 𝟔
 Interchange the position
of x and y
𝒙 =
𝟕 + 𝒚
𝟐𝒚 − 𝟔
 Simplify the rational
expression
𝒙 𝟐𝒚 − 𝟔 = 𝟕 + 𝒚
2𝒙𝒚 − 𝟔𝒙 = 𝟕 + 𝒚
 Solve for y in terms of x 𝟐𝒙𝒚 − 𝒚 = 𝟕 + 𝟔𝒙
𝒚(𝟐𝒙 − 𝟏)
𝟐𝒙 − 𝟏
=
𝟕 + 𝟔𝒙
𝟐𝒙 − 𝟏
𝒚 =
−𝟕 + 𝟔𝒙
𝟐𝒙 − 𝟏
 Equate the
denominator
to 0.
𝟐𝒙 − 𝟏 = 𝟎
The range of 𝒇(𝒙) is the set of all
real numbers except
𝟏
𝟐
.
𝑹: 𝒚 𝒚 ∈ ℝ, 𝒚 ≠
𝟏
𝟐
𝒚 𝟐𝒙 − 𝟏 = 𝟕 + 𝟔𝒙
𝟐𝒙 = 𝟏
𝒙 =
𝟏
𝟐

Vertical and
Horizontal
Asymptotes of
Rational Functions

They are the restrictions on the x
– 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

Let n be the degree of the numerator and m
be the degree of denominator:
• If 𝒏 < 𝒎, 𝒚 = 𝟎.
• If 𝒏 = 𝒎, 𝒚 =
𝒂
𝒃
, where 𝒂 is the leading
coefficient of the numerator and 𝒃 is the
leading coefficient of the denominator.
• If 𝒏 > 𝒎 , there is no horizontal
asymptote.
Finding the Horizontal Asymptotes
of Rational Functions

Find the Degree of Polynomial.
𝟓𝒙 𝟏
𝑫𝒆𝒈𝒓𝒆𝒆
𝒙 − 𝟒 𝟏
𝟐𝒙𝟑
− 𝒙 − 𝟒 𝟑

Find the Degree of Polynomial.
𝒙𝟐
− 𝟐𝒙𝟓
− 𝒙 𝟓
𝑫𝒆𝒈𝒓𝒆𝒆
𝒚𝟐
− 𝒚 + 𝟏 𝟐
𝟗 + 𝟐𝒙 − 𝒙𝟑 𝟑

EXAMPLE 1:
𝒇 𝒙 =
𝟑
𝒙 − 𝟓

𝒇 𝒙 =
𝟑
𝒙 − 𝟓
To find the vertical
asymptote:
𝒙 − 𝟓 = 𝟎
𝒙 = 𝟓
Focus on the
denominator
The vertical asymptote
is 𝒙 = 𝟓.
EXAMPLE 1:

𝒇 𝒙 =
𝟑
𝒙 − 𝟓
To find the horizontal
asymptote:
𝒏 < 𝒎
Focus on the degree
of the numerator
and denominator
The horizontal asymptote
is 𝒚 = 𝟎.
0
1
EXAMPLE 1:

𝒇 𝒙 =
𝟒𝒙 − 𝟐
𝒙 + 𝟐
EXAMPLE 2:

𝒇 𝒙 =
𝟒𝒙 − 𝟐
𝒙 + 𝟐
To find the vertical
asymptote:
𝒙 + 𝟐 = 𝟎
𝒙 = −𝟐
Focus on the
denominator
The vertical asymptote
is 𝒙 = −𝟐.
EXAMPLE 2:

𝒇 𝒙 =
𝟒𝒙 − 𝟐
𝒙 + 𝟐
To find the horizontal
asymptote:
𝒏 = 𝒎
Focus on the degree
of the numerator
and denominator
The horizontal
asymptote is 𝒚 = 𝟒.
1
1
EXAMPLE 2:
𝒚 =
𝒂
𝒃
=
𝟒
𝟏
= 𝟒
a is the leading coefficient of 4x
b is the leading coefficient of x

𝒇 𝒙 =
𝟑𝒙 + 𝟒
𝟐𝒙𝟐 + 𝟑𝒙 + 𝟏
EXAMPLE 3:

𝒇 𝒙 =
𝟑𝒙 + 𝟒
𝟐𝒙𝟐 + 𝟑𝒙 + 𝟏
To find the vertical asymptote:
𝟐𝒙𝟐 + 𝟑𝒙 + 𝟏 = 𝟎
Focus on the
denominator
The vertical
asymptote are 𝒙 = −
𝟏
𝟐
and 𝒙 = −𝟏.
EXAMPLE 3:
𝟐𝒙 + 𝟏 𝒙 + 𝟏 = 𝟎
𝟐𝒙 + 𝟏 = 𝟎 𝒙 + 𝟏 = 𝟎
𝟐𝒙 = −𝟏
𝒙 = −
𝟏
𝟐
𝒙 = −𝟏

1
2
EXAMPLE 3:
𝒇 𝒙 =
𝟑𝒙 + 𝟒
𝟐𝒙𝟐 + 𝟑𝒙 + 𝟏
To find the horizontal
asymptote:
𝒏 < 𝒎
Focus on the degree
of the numerator
and denominator
The horizontal asymptote
is 𝒚 = 𝟎.

𝒇 𝒙 =
𝟒𝒙𝟑
− 𝟏
𝒙𝟐 + 𝟒𝒙 − 𝟓
EXAMPLE 4:

𝒇 𝒙 =
𝟒𝒙𝟑
− 𝟏
𝒙𝟐 + 𝟒𝒙 − 𝟓
To find the vertical asymptote:
𝒙𝟐 + 𝟐𝒙 − 𝟓 = 𝟎
Focus on the
denominator
The vertical
asymptote are 𝒙 = −𝟓
and 𝒙 = 𝟏.
EXAMPLE 4:
𝒙 + 𝟓 𝒙 − 𝟏 = 𝟎
𝒙 + 𝟓 = 𝟎 𝒙 − 𝟏 = 𝟎
𝒙 = −𝟓 𝒙 = 𝟏
𝒙 = −𝟓

3
2
EXAMPLE 4:
To find the horizontal
asymptote:
𝒏 > 𝒎
Focus on the degree
of the numerator
and denominator
The rational function has
no horizontal asymptote.
𝒇 𝒙 =
𝟒𝒙𝟑
− 𝟏
𝒙𝟐 + 𝟒𝒙 − 𝟓

Zeros of
Rational
Function

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).

𝒇 𝒙 =
𝒙𝟐
− 𝟒𝒙
𝒙 + 𝟏
EXAMPLE 1:

𝒇 𝒙 =
𝒙𝟐
− 𝟒𝒙
𝒙 + 𝟏
 Factor the numerator and
denominator
EXAMPLE 1:
𝒇(𝒙) =
𝒙(𝒙 − 𝟒)
𝒙 + 𝟏
 Identify the restrictions. 𝒙 + 𝟏 = 𝟎
𝒙 = −𝟏
 Identify the values of x that
will make the numerator
equal to zero.
𝒙 𝒙 − 𝟒 = 𝟎
𝒙 = 𝟎 𝒙 − 𝟒 = 𝟎
𝒙 = 𝟒
 Identify the zeroes of f(x).
𝒙 = 𝟎 𝒙 = 𝟒

𝒇 𝒙 =
(𝒙 − 𝟒)(𝒙 + 𝟐)
(𝒙 − 𝟑)(𝒙 − 𝟏)
EXAMPLE 2:

 Factor the numerator and
denominator
EXAMPLE 2:
𝒇(𝒙) =
(𝒙 − 𝟒)(𝒙 + 𝟐)
(𝒙 − 𝟑)(𝒙 − 𝟏)
 Identify the restrictions. 𝒙 − 𝟑 𝒙 − 𝟏 = 𝟎
𝒙 − 𝟑 = 𝟎
 Identify the values of x
that will make the
numerator equal to
zero.
𝒙 − 𝟒 𝒙 + 𝟐 = 𝟎
𝒙 − 𝟒 = 𝟎 𝒙 + 𝟐 = 𝟎
𝒙 = −𝟐
 Identify the zeroes of f(x).
𝒙 = 𝟒 𝒙 = −𝟐
𝒇 𝒙 =
(𝒙 − 𝟒)(𝒙 + 𝟐)
(𝒙 − 𝟑)(𝒙 − 𝟏)
𝒙 = 𝟑
𝒙 − 𝟏 = 𝟎
𝒙 = 𝟏
𝒙 = 𝟒

𝒇 𝒙 =
𝒙𝟐
+ 𝟓𝒙 + 𝟒
𝒙𝟐 − 𝟐𝒙 − 𝟑
EXAMPLE 3:

 Factor the numerator and
denominator
EXAMPLE 3:
𝒇(𝒙) =
(𝒙 + 𝟏)(𝒙 + 𝟒)
(𝒙 − 𝟑)(𝒙 + 𝟏)
 Identify the restrictions. 𝒙 − 𝟑 𝒙 + 𝟏 = 𝟎
𝒙 − 𝟑 = 𝟎
 Identify the values of x
that will make the
numerator equal to
zero.
𝒙 + 𝟏 𝒙 + 𝟒 = 𝟎
𝒙 + 𝟏 = 𝟎 𝒙 + 𝟒 = 𝟎
𝒙 = −𝟒
 Identify the zeroes of f(x).
𝒙 = −𝟒
𝒙 = 𝟑
𝒙 + 𝟏 = 𝟎
𝒙 = −𝟏
𝒙 = −𝟏
𝒇 𝒙 =
𝒙𝟐
+ 𝟓𝒙 + 𝟒
𝒙𝟐 − 𝟐𝒙 − 𝟑

Intercepts of
Rational
Functions

 Intercepts are x and y – coordinates of
the points at which a graph crosses the
x-axis or y-axis, respectively.
 y-intercept is the y-coordinate of the
point where the graph crosses the y-
axis.
 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.

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.

𝒇 𝒙 =
𝒙 + 𝟒
𝒙 − 𝟐
EXAMPLE 1:

𝒇 𝒙 =
𝒙 + 𝟒
𝒙 − 𝟐
EXAMPLE 1:
y - intercept
𝒇(𝒙) =
𝒙 + 𝟒
𝒙 − 𝟐
𝒙 = 𝟎
𝒇(𝒙) =
𝟎 + 𝟒
𝟎 − 𝟐
𝒇(𝒙) =
𝟒
−𝟐
𝒇(𝒙) = −𝟐
x - intercept
𝒇(𝒙) =
𝒙 + 𝟒
𝒙 − 𝟐
𝒚 = 𝟎
𝟎 =
𝒙 + 𝟒
𝒙 − 𝟐
𝟎 𝒙 − 𝟐 = 𝒙 + 𝟒
𝟎 = 𝒙 + 𝟒
−𝟒 = 𝒙

𝒇 𝒙 =
𝒙𝟐
+ 𝟓𝒙 + 𝟒
𝒙𝟐 − 𝟐𝒙 − 𝟑
EXAMPLE 2:

EXAMPLE 1:
y - intercept
𝒙 = 𝟎
𝒇 𝒙 =
𝟎 + 𝟒
𝟎 − 𝟑
x - intercept
𝒚 = 𝟎
𝟎 = 𝒙 + 𝟒
−𝟒 = 𝒙
𝒇 𝒙 =
𝒙𝟐
+ 𝟓𝒙 + 𝟒
𝒙𝟐 − 𝟐𝒙 − 𝟑
𝒇 𝒙 =
(𝒙 + 𝟏)(𝒙 + 𝟒)
(𝒙 + 𝟏)(𝒙 − 𝟑)
𝒇 𝒙 =
𝒙 + 𝟒
𝒙 − 𝟑
𝒇 𝒙 = −
𝟒
𝟑
𝒇 𝒙 =
𝒙 + 𝟒
𝒙 − 𝟑
𝟎 =
𝒙 + 𝟒
𝒙 − 𝟑
𝟎(𝒙 − 𝟑) = 𝒙 + 𝟒

Find each of the following:
a) Intercepts
1)𝑓 𝑥 =
𝑥
𝑥+4
2) 𝑓 𝑥 =
𝑥 −7
𝑥 −5
3) 𝑓 𝑥 =
𝑥2−5𝑥−14
𝑥2−4