The document discusses finding the domain, range, zeros, intercepts, and asymptotes of rational functions. It provides examples of finding: 1) The domain by identifying values that make the denominator equal to zero. 2) The range by changing the function to be in terms of y and solving for y. 3) Vertical asymptotes by making the denominator equal to zero. 4) Horizontal asymptotes based on the degree of the numerator and denominator. 5) Zeros by factoring and setting the numerator equal to zero.

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

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

3.  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

5. 𝒇 𝒙 =
𝟐
𝒙 − 𝟑
EXAMPLE 1:

6. 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

7. 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

8. 𝒇 𝒙 =
𝒙 − 𝟓
𝒙 + 𝟐
EXAMPLE 2:

9. 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

10. 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

11. 𝒇 𝒙 =
𝟕 + 𝒙
𝟐𝒙 − 𝟔
EXAMPLE 3:

12. 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

13. 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

14. Vertical and
Horizontal
Asymptotes of
Rational Functions

15. 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

16. 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

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

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

20. EXAMPLE 1:
𝒇 𝒙 =
𝟑
𝒙 − 𝟓

21. 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:

22. 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:

23. 𝒇 𝒙 =
𝟒𝒙 − 𝟐
𝒙 + 𝟐
EXAMPLE 2:

24. 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:

25. 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

26. 𝒇 𝒙 =
𝟑𝒙 + 𝟒
𝟐𝒙𝟐 + 𝟑𝒙 + 𝟏
EXAMPLE 3:

27. 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

28. 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.

29. 𝒇 𝒙 =
𝟒𝒙𝟑
− 𝟏
𝒙𝟐 + 𝟒𝒙 − 𝟓
EXAMPLE 4:

30. 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

31. 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

32. Zeros of
Rational
Function

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).

35. 𝒇 𝒙 =
𝒙𝟐
− 𝟒𝒙
𝒙 + 𝟏
EXAMPLE 1:

36. 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

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

38.  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

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

40.  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

41. Intercepts of
Rational
Functions

42.  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.

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.

45. 𝒇 𝒙 =
𝒙 + 𝟒
𝒙 − 𝟐
EXAMPLE 1:

46. 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

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

48. 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

50. 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