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Unformatted text preview: 1.7 Combinations of Functions; Composite Functions 1. Find the domain of a function. To ﬁnd the domain of a function deﬁned by an equation: 1. Start with the domain as the set of real numbers. 2. If the equation has a denominator, exclude any numbers that give a zero denominator. 3. If the equation has a radical of even index, exclude any numbers that cause the expression inside the radical to be negative. Note: When the domain is not speciﬁed, in this class we assume that the domain is the set of all real numbers that, as inputs, produce real number outputs.
Example: Find the domain of f(x) = 1 1 x—2 x+3l .0 2. Combine functions using the algebra of functions, specifying domains. Theﬁlégeimt of ﬂatworm31ml, ﬁiﬂereﬁee; Premier, am} {Examiner/zit 9f Fmeﬁﬁm Let f and g ”be two ﬁmetioasﬂfhe sum f + g.i the ﬂiﬂermoe f w gt {he pmﬂuel
ﬁg and the garment: wig: are innitﬁrms whose domains are the sea of all real numbers common in the domains of f and. gf ﬁfﬂ ﬁg}? deﬁned as follows: L Emil; if 4‘ ﬁll“ a fix}. “l” m} 2; Swimmer {if "r sl(xl *4 fix?) M six} .3. Product: GEEK} ”“3 fill”ﬁx! 4s Quotient, (g){x} ...,. gilt}? pmwéed gm} sh EL The domain of f+g is the set of all real numbers that
are common to the domain of f and the domain of g.
Thus we must ﬁnd the domains of f and g before ﬁnding their intersection.
The domain of f  g is the set of all real numbers that are common to the domain of f and the domain of g.
Thus we must ﬁnd the domains of f and g before
ﬁnding their intersection ELM
f(x)=:2xj126 and g(x)=:2x__146, ﬁnd f+g, f—g, fg,§. Determine the
domain of each. ex '+
'F(X)+3CX) :' iii]: +573? x/ Xié/ “if 3x+a+ 5x4? xl—Ié %X a Xlﬂlé 3. Form composite functions: Given two functions f and g, the composite function,
denoted by f o g (read 63“)" composed with g“).
is defined by (feg >x>< ﬂux) The domain off 0 g is the set of all numbers x
in the domain of g such that
g(x) is in the domain of f, Bomain of g R'
., an e f . ~ .. g 1W9?! _ﬁ_ £1 Dwain U? f Range eff , w; M ay 4 . . _‘ i=5; Dumam ef fa g Range 0f 1‘0 g 1. g(x) must be defined so any x not in the domain of g must be excluded. 2. f ( g(x)) must be defined so any x for which g(x) is not in the domain of f must.
be excluded We call f(g(x)) the composition of the function f with g, or a
composite function. This composite function is written f o g ...
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 Fall '11
 BlisinHestiyas

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