1.7, page 209
Combinations of
Functions; Composite
Functions
Objectives
Find
the domain of a function.
Combine functions using algebra.
Form composite functions.
Determine domains for composite
functions.
Using basic algebraic functions, what
limitations are there when working with
real numbers?

A) You canNOT divide by zero.
Any values that would result in a zero
denominator are NOT allowed, therefore the
domain of the function (possible x values)
would be limited.
B) You canNOT take the square root (or
any even root) of a negative number.
Any values that would result in negatives
under an even radical (such as square roots)
result in a domain restriction.

Reminder: Domain Restrictions
(WRITE IT DOWN!)
For FRACTIONS:
 No zero in denominator!
ex.
7
is undefinedif x  3
x 3
For EVEN ROOTS:
 No negative under radical!
ex.
x 2 ,
4
x 2
x  2 is undefinedif x  2  0
Check Point 1, page 211
Find the domain of each function.
a ) f ( x)  x  3 x  17
2
5x
b) g ( x )  2
x  49
c) h( x)  9 x  27
The Algebra of Functions, pg 213
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f  g)(x) = f(x)  g(x)
(f / g)(x) = f(x) / g(x), g(x)  0
Finding Domain of
f(x) plus, minus, times, or divided by g(x)
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f  g)(x) = f(x)  g(x)
Domain  [domain of f ( x)] [domain of g ( x )]
(f / g)(x) = f(x) / g(x)
Domain  [domain of f ( x)] [domain of g ( x)], where g ( x)  0
See Example 2, page 213.
Check Point 2, page 214.

a)
b)
Let f(x) = x – 5 and g(x) = x2 – 1. Find
(f + g)(x)
(f - g)(x)
c)
(fg)(x)
d)
(f/g)(x)
See Example 3, page 214.
Check Point 3
Let f ( x)  x  3 and g ( x)  x  1. Find
a) ( f  g )( x)
b) the domain of f  g
Given two functions f and g, the
composite function, denoted by
f o g (read as “f of g of x”)
is defined by
f
g   x   f  g  x 
Composition of Functions
See Example 4, page 217.
Check Point 4
Given f(x)=5x+6 and g(x)=2x2 – x – 1,
find
a) f(g(x))
b) g(f(x))
Domain of f o g
Note: Finding the domain of f(g(x)) is
NOT the same as finding f(x) + g(x)
The domain of f o g is the set of
all numbers x in the domain of
g such that g( x) is in the domain
of f.
How to find the domain of a
composite function
1.
2.
3.
Find the domain of the function that is
being substituted (Input Function)
into the other function.
Find the domain of the resulting
function (Output Function).
The domain of the composite function
is the intersection of the domains
found above.
See Example 5, page 218.
 Check
Point 5
4
1
Let f ( x) 
and g ( x)  . Find
x2
x
a) ( f g )( x)
b) the domain of f g
Extra Domain Example


Find the domain
x2
2
x  5x  6
There are x’s under an even radical AND x’s in
the denominator, so we must consider both of
these as possible limitations to our domain.
x  2  0 and x  5x  6  0
2
Extra Example: Operations with Functions
Given that f(x) = x2 - 4 and g(x) = x + 2, find:
a)
(f+g)(x) =
b)
(f-g)(x) =
c)
(fg)(x) =
d)
(f/g)(x) =
Remember: f(x) = x2 - 4 and g(x) = x + 2.
Now, let’s find the domain of each answer.
a)
(f+g)(x) = x2 + x - 2
b)
(f-g)(x) = x2 – x - 6
c)
(fg)(x) = x3 – 2x2 – 4x - 8
d)
(f/g)(x) = x – 2
Extra Example: Composition
Given f(x)=2x – 5 & g(x)=x2 – 3x + 8,
find
a) (f◦g)(x) and (g◦f)(x)
b) (f◦g)(7) and (g◦f)(7)
c) What is the domain of these composite
functions?