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Study Notes

Rational Functions

October 25, 2022

A rational expression is a ratio of two polynomials
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Rational Function
f(x) =

𝑝(𝑥)
𝑞 (𝑥)

where p(x) and q(x) are polynomial functions, with q(x) ≠ 0
Examples

𝑥 2 +2𝑥−3

𝑥+1

3𝑥+7

2𝑥+6

4𝑥 2 +6𝑥−4

𝑥−5

The domain of a rational function is the set of all real numbers except those that result to the
denominator becoming zero
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This is the case
because division by zero is undefined
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Determining the Domain of Rational Functions
Find the domain of the following functions
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𝑥+5
𝑥−2

b
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1
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Solve for x
x-2+2 = 0+2
x=2
3
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Since the value of the denominator is zero when the value of x is 2, then
the domain of the function is the set of all real numbers except 2
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In set builder notation, {x | x ≠ 2}
Solution b
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Equate the denominator to zero
x2 -4 = 0
2
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Interpret the result
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In interval notation, (-ꝏ, -2) Ս (-2, 2) Ս (2, ꝏ)
In set builder notation {x | x ≠2,-2}
1

Reymon T
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Simplifying Rational Functions
To express a rational Function to its lowest term or simplest form, we factor out the numerator and
the denominator then divide out the common factors
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a
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𝑥 2 −16
𝑥 2 −7𝑥+12

c
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Factor the numerator and the denominator
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In this case x+2
3(𝑥 + 2)
(𝑥 + 2)(𝑥 + 5)
𝟑
𝒙+𝟓
Solution b
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(𝑥 − 4)(𝑥 + 4)
(𝑥 − 4)(𝑥 − 3)
Divide out all common factors
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Factor the numerator and the denominator
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In this case x-5
(𝑥−5)(𝑥+3)
3𝑥(𝑥−5)(𝑥+2)
2

Reymon T

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