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Chapter
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12: Rational Functions
12.1 SIMPLIFYING RATIONAL
FUNCTIONS
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Rational Function
P( x)
f ( x) =
Q( x)
ƒ Function that can be written as a ratio
ƒ Where P(x) and Q(x) are polynomials
ƒ As long as Q(x) is not zero
Reducing fractions
ƒ Recall that any number divided by itself is
1
ƒ Recall that fraction bar is a grouping
symbol
b l
ƒ Can reduce fraction if the same FACTOR
appears above and below the fraction bar
ƒ If the same term appears
appears, it is part of a
sum, not a factor, and cannot be reduced
Reducing fractions
ƒ Like this
12 4 ⋅ 3
4 3
3
3
=
= ⋅
= 1⋅
=
20 4 ⋅ 5
4 5
5
5
5+ 4
9
=
14 + 4 18
9 1
= ⋅
9 2
ƒ Not like this
5+ 4
5
≠
14 + 4 14
=
1
2
Reducing Rational Functions
ƒ Factor when possible
ƒ Reduce factors common to numerator and
denominator
ƒ Remember that if all factors cancel from
either or both,, 1 is always
y another factor
present
Reduce this function
ƒ Factor the polynomials
ƒ Cancel factors matching above and below
fraction bar: because they equal 1
x 2 − 4 = ( x − 2 )( x + 2) = x − 2 ⋅ ( x + 2) = x + 2
f ( x) =
(x − 2)
x−2
x−2
Evaluate functions
ƒ f(3)
ƒ =5
5
ƒ f(-4)
ƒ = —2
x −4
f ( x) =
x−2
2
f ( x) = x + 2
ƒ f(2)
ƒ Does not exist
Domain of functions
ƒ Some are all real numbers
ƒ Some are all real numbers except where
denominator is zero
ƒ Often this is not apparent in the simplified
function
ƒ Sometimes it appears in graph as vertical
asymptote: a gap in the function
ƒ This is an EXCLUDED VALUE of the
rational
ti
l expression
i
Reduce this function
ƒ Factor the polynomials
ƒ Cancel factors matching above and below
fraction bar: because they equal 1
5 x + 15
f ( x) = 2
x + 5x + 6
5( x + 3)
=
(x + 2)(x + 3)
5
=
x+2
Reduce this function
ƒ Factor the polynomials
ƒ Cancel factors matching above and below
fraction bar: because they equal 1
2 x 2 − 6 x − 20 2(x 2 − 3 x − 10 )
f ( x) =
=
2
2
2 x − 50
2x
2(x − 25)
(
x + 2)
=
( x + 5)
2( x − 5)( x + 2 )
=
2( x − 5)( x + 5)