3. DOMAIN AND RANGE OF RATIONAL
FUNCTION
The Domain of a rational function 𝐟(𝐱)
𝑵(𝒙)
𝑫 (𝒙)
is all the values of 𝒙 that 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.
4. 𝐸𝑥𝑎𝑚𝑝𝑙𝑒𝑠:
𝑓 𝑥 =
2
𝑥 − 3
To find the domain:
Equate the denominator by
zero
𝑥 − 3 = 0
𝒙 = 𝟑
The domain of 𝑓(𝑥) is the set
of all real numbers except 3
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑦 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑥:
𝑦 =
2
𝑥 − 3
𝑥 =
2
𝑦 − 3 Interchange their place
𝑥 =
2
𝑦 − 3
Cross multiply
𝑥 𝑦 − 3 = 2
𝑥𝑦 − 3𝑥 = 2
𝑥𝑦 = 2 + 3𝑥
𝑥𝑦
𝑥
=
2 + 3𝑥
𝑥
𝑦 =
2 + 3𝑥
𝑥 Equate the denominator by zero
𝒙 = 𝟎 The range of 𝑓(𝑥) is the set
of all real numbers except 0
5. 𝐸𝑥𝑎𝑚𝑝𝑙𝑒𝑠:
𝑓 𝑥 =
𝑥 − 5
𝑥 + 2
𝑥 + 2 = 0
𝒙 = −𝟐
The domain of 𝑓(𝑥) is the set
of all real numbers except -2
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑦 𝑖𝑛 𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑥:
𝑦 =
𝑥 − 5
𝑥 + 2
𝑦 =
𝑥 − 5
𝑥 + 2
Interchange their place
𝑥 =
𝑦 − 5
𝑦 + 2
Cross multiply
𝑥 𝑦 + 2 = 𝑦 − 5
𝑥𝑦 + 2𝑥 = 𝑦 − 5
𝑥𝑦 − 𝑦 = −5 − 2𝑥
𝑦 =
−5 − 2𝑥
𝑥 − 1 Equate the denominator by zero
𝒙 = 𝟏 The range of 𝑓(𝑥) is the set
of all real numbers except 1
𝑠𝑜𝑙𝑣𝑒 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 𝑜𝑓 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑦(𝑥 − 1) = −5 − 2𝑥
𝑦(𝑥 − 1)
𝑥 − 1
=
−5 − 2𝑥
𝑥 − 1
𝑥 − 1 = 0
6. 𝐸𝑥𝑎𝑚𝑝𝑙𝑒𝑠:
𝑓 𝑥 =
(𝑥 − 4)(𝑥 + 2)
(𝑥 − 3)(𝑥 − 1)
𝑥 − 3 = 0
The domain of 𝑓(𝑥) is the set
of all real numbers except 3
and 1
𝑈𝑠𝑖𝑛𝑔 𝑐𝑜𝑛𝑐𝑒𝑝𝑡 𝑜𝑓 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒
The range of 𝑓(𝑥) is the set
of all real numbers except 1
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑒𝑞𝑢𝑎𝑡𝑒 𝑡ℎ𝑒𝑚 𝑜𝑛𝑒 𝑏𝑦 𝑜𝑛𝑒
𝒙 = 𝟑
𝑥 − 1 = 0
𝒙 = 𝟏
𝑛 < 𝑚 , 𝑦 = 0
𝑛 = 𝑚 , 𝑦 =
𝑎
𝑏
𝑛 > 𝑚,
𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒
𝑓 𝑥 =
(𝑥 − 4)(𝑥 + 2)
(𝑥 − 3)(𝑥 − 1)
𝑓 𝑥 =
𝑥2 − 2𝑥 − 8
𝑥2 − 4𝑥 + 3
𝒚 =
𝒂
𝒃
=
𝟏
𝟏
= 𝟏
7. 𝐸𝑥𝑎𝑚𝑝𝑙𝑒𝑠:
𝑓 𝑥 =
3𝑥 − 9
(𝑥 − 3)(𝑥 + 2)
𝑓 𝑥 =
3𝑥 − 9
𝑥2 − 𝑥 − 6
factor
𝒇 𝒙 =
𝟑𝒙 − 𝟗
𝒙 𝟐 − 𝒙 − 𝟔
𝑥 − 3 = 0 𝑥 + 2 = 0
𝒙 = 𝟑 𝒙 = −𝟐
The domain of 𝑓(𝑥) is the set of all
real numbers except 3 and -2
𝑈𝑠𝑖𝑛𝑔 𝑐𝑜𝑛𝑐𝑒𝑝𝑡 𝑜𝑓 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒
𝑛 < 𝑚 , 𝑦 = 0
𝑛 = 𝑚 , 𝑦 =
𝑎
𝑏
𝑛 > 𝑚,
𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒
𝒇 𝒙 =
𝟑𝒙 − 𝟗
𝒙 𝟐 − 𝒙 − 𝟔
𝒚 = 𝟎
The range of 𝑓(𝑥) is the set
of all real numbers except 0
8. 𝐸𝑥𝑎𝑚𝑝𝑙𝑒𝑠:
𝒇 𝒙 =
𝟑𝒙 𝟐 − 𝟖𝒙 − 𝟑
𝟐𝒙 𝟐 + 𝟕𝒙 − 𝟒
The domain of 𝑓(𝑥) is the set of all
real numbers except
1
2
and −4
𝑈𝑠𝑖𝑛𝑔 𝑐𝑜𝑛𝑐𝑒𝑝𝑡 𝑜𝑓 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒
𝑛 < 𝑚 , 𝑦 = 0
𝑛 = 𝑚 , 𝑦 =
𝑎
𝑏
𝑛 > 𝑚,
𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒
The range of 𝑓(𝑥) is the set
of all real numbers except
3
2
𝑓 𝑥 =
3𝑥2 − 8𝑥 − 3
2𝑥2 + 7𝑥 − 4
𝑓 𝑥 =
(3𝑥 + 1)(𝑥 − 4)
(2𝑥 − 1)(𝑥 + 4)
factor
2𝑥 − 1 = 0 𝑥 + 4 = 0
2𝑥 = 1 𝒙 = −𝟒
𝒙 =
𝟏
𝟐
𝒇 𝒙 =
𝟑𝒙 𝟐 − 𝟖𝒙 − 𝟑
𝟐𝒙 𝟐 + 𝟕𝒙 − 𝟒
𝑓 𝑥 =
3𝑥2 − 8𝑥 − 3
2𝑥2 + 7𝑥 − 4
𝒚 =
𝒂
𝒃
=
𝟑
𝟐
