Math 1314
2.6 Combinations of Functions-Composite Functions
Section 2.6 Notes
Domain of Functions
Finding the Domain of a Function
Definition: Given a function f:X→Y such that y = f(x). The set X is the domain of f then for every x in X, f(x)
is defined in Y.
Remark: The implied is the set of all real numbers for which the expression of function is defined.
The question one must ask when finding the domain is “where is this function NOT defined?”
Therefore:
Given function y =f(x), to find the domain of f:
- If f(x) is a polynomial, then the domain of f is all real numbers or (- ∞, ∞).
-If f(x) is a rational expression, then the domain of f is all real numbers except the value(s) of x that make(s)
the denominator equal to zero.
-If f(x) is a radical expression with the even root, then the domain of f is the set of all real numbers that make
the radicand non-negative.
-If f(x) is a radical expression with the odd root, then the domain of f is the set of all real numbers.
Example: Find the domain of each function. Then express the domain in interval notation.
1.
f ( x) 
2x
x  5x  6
2
1
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Math 1314
Section 2.6 Notes
2.
f ( x) 
x
9  x2
3.
f ( x) 
x5
8 x  2 x 2  3x
3
2
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Math 1314
Section 2.6 Notes
x 5
x 2  25
4.
f ( x) 
5.
f ( x)  x 3  5x 2  7
6.
f ( x)  x  5
7.
f ( x)  4  5 x
3
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Math 1314
8. f ( x )  3x  7
9.
Section 2.6 Notes
f ( x)  24  12 x  7
10. f ( x )  3 x  2
4
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Math 1314
Section 2.6 Notes
1
3x  1
11. f ( x ) 
4
12. f ( x) 
x2
x  16
2
5
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Math 1314
Operations on Functions
Section 2.6 Notes
Sum: (f + g)(x) = f(x) + g(x)
Difference: (f – g)(x) = f(x) – g(x)
Product: (f ∙ g)(x) = f(x) ∙ g(x)
Quotient: (f ÷ g)(x) = f(x) ÷ g(x)
Examples: For each pair of functions f(x), g(x), find:
1. f(x) = 2x – 4,
g(x) = x2 + 3
a/ (f + g)(5)
b/ (f + g)(x)
c/ (f – g)(x)
d/ (fg)(x)
e/ g  f (x) 
2. f(x) = x2 + 2x + 3,
g(x) = x – 1
6
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Math 1314
Section 2.6 Notes
a/ (f + g)(5)
b/ (f + g)(x)
c/ (f – g)(x)
d/ (fg)(x)
e/ f  g (x) 
7
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Math 1314
Composite Functions
Section 2.6 Notes
Composite functions are functions that are formed from two functions f(x) and g(x) in which the output or result
of one of the functions is used as the input to the other function. Notationally, we express composite functions
as ( f  g )x  or f g x  .
In this case the result or output from g becomes the input to f.
( f  g )x  or f g  x  is read “the composition f with g” or “f of g of x”.
Generally, composite functions can be formed from more than two functions i.e. the composite function formed
from three functions f(x), g(x) and h(x) is ( f  g  h )x  or f g h( x )  .
Example: Given f ( x) 
2
3
and g ( x)  . Find the following:
x2
x
a/  f  g (1)
b/  g  f (x )
8
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Math 1314
Section 2.6 Notes
Examples: Find the composite functions ( f  g ) x  and ( g  f )( x ) then use the results to determine whether the
composition is commutative.
1. f x   x 3 and g x   x  2
f x   2 x  4 and g  x   x  1
9
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Math 1314
2. f  x  
Section 2.6 Notes
x
1
and g  x  
x 1
x
2
10
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Math 1314
Section 2.6 Notes
11
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