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Review -- Graph the basic function as a solid line
and the transformation as a dashed line.

f(x) = x2,
g(x) = (x-1)2 + 2
y
10
x
-10
-10
10
Graph the basic function as a solid line and the
transformation as a dashed line.

f(x) = |x|,
g(x) = -|x + 3| - 1
y
10
x
-10
-10
10
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?
1) You canNOT divide by ________.
 Any values that would result in a zero
denominator are NOT allowed, therefore the
domain of the function (possible x values)
would be limited.
2) You canNOT take the square root (or
any even root) of a ________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!
7
ex.
 undefined
0
For EVEN ROOTS:
 No negative under radical!
ex.
x 2 ,
4
x 2
Find the domain of each function.
a ) f ( x)  x  3x  17
2
5x
b) g ( x )  2
x  49
c) h( x)  9 x  27
The Algebra of Functions
(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
Examples

Let f(x) = x – 5 and g(x) = x2 – 1. Find
d) (f + g)(x)
e) (f - g)(x)
f) (fg)(x)
g) (f/g)(x)
Examples
Let f ( x)  x  3 and g ( x)  x  1. Find
h) ( f  g )( x)
i) the domain of f  g
Given two functions f and g, the
________ function , denoted by
f o g (read as “f composed with
g”), is defined by
f
g   x   f  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.
Composition of Functions
Given f(x)=5x+6 and g(x)=2x2 – x – 1, find
j) (f◦g)(x)
k) (g◦f)(x)
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 ______function
(Output Function).
The domain of the composite function
is the __________ of the domains
found above.
Example
4
1
Let f ( x) 
and g ( x)  . Find
x2
x
l) ( f g )( x)
m) the domain of f
g
Extra Example 1


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 2: Operations with Functions
Given that f(x) = x2 - 4 and g(x) = x + 2, find each
and the resulting domain.
a)
(f+g)(x) =
b)
(f-g)(x) =
c)
(fg)(x) =
d)
(f/g)(x) =
Extra Example 3: 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 4: Composition
Given f(x)=2x – 5 & g(x)=x2 – 3x + 8,
find
(f◦g)(x) and (g◦f)(x)
(f◦g)(7) and (g◦f)(7).