Transcript composition-of-functions

Daily Check
Simplify:
2 5
2
(2
x
y
)
1)
3x 2 y 5 z 4
2)
6 x3 yz 6
Math II
UNIT QUESTION: What methods
can be used to find the inverse of a
function?
Standard: MM2A2, MM2A5
Today’s Question:
How do you find the composite of
two functions and the resulting
domain?
Standard: MM2A5.d
4.2
Composition of Functions
Objective
 To form and evaluate composite functions.
 To determine the domain for composite functions.
Composition of functions
 Composition of functions is the successive
application of the functions in a specific order.
 Given two functions f and g, the composite function
f g is defined by  f g  x   f  g  x   and is read
“f of g of x.”
 The domain of f g is the set of elements x in the
domain of g such that g(x) is in the domain of f.
 Another way to say that is to say that “the range of function
g must be in the domain of function f.”
A composite function
f g
x
g
g(x)
domain of g
f
range of f
f(g(x))
range of g
domain of f
A different way to look at it…
f  g  x 
x
f
g
Function
Machine
gFunction
x
Machine
Example 1
 Evaluate  f g  x  and  g f  x  :
f
x  x  3
g
 x   2x
2


gf  fgxx  2 2
 xx 2 31  31
1
2


 22 xx2246 x  9  1
 2 x 2  12 x  18  1
f
g  x   2x 2  4
 g f  x   2x 2  12x  17
You can see that function composition is not commutative!
Example 2
 Evaluate  f g  x  and  g f  x  :

f  x   2x

g x  x
3
1
      2x
fg gf x x  2x
1 3
 2 x 3
2
 3
x
2
 f g  x   3
x
1
 g f  x   3
2x

3 1
1
3
2x
Again, not the same function. What is the domain???
Example 3
 Find the domain of  f g  x  and  g f  x :

f x  x 1
g
x 
f
x
g  x   x  1
Df
g
 x : x  0
(Since a radicand can’t be negative in the set of real numbers,
f be
x  greater
 x than
1 or
Dgequal
x  1
xgmust
f  x
to: zero.)
(Since a radicand can’t be negative in the set of real numbers,
x – 1 must be greater than or equal to zero.)
Your turn
 Evaluate  f g  x  and  g f  x  :

f  x   3x 2

g x  x  5
Example 4
 Find the indicated values for the following functions
if:

f  x   2x  3
2
g
x

x
1
 

f (g (1))
f (g (4))
g (f (2))
g (g (2))
Example 5
 The number of bicycle helmets produced in a factory
each day is a function of the number of hours (t) the
assembly line is in operation that day and is given by
n = P(t) = 75t – 2t2.
 The cost C of producing the helmets is a function of
the number of helmets produced and is given by
C(n) = 7n +1000.
Determine a function that gives the cost of producing the
helmets in terms of the number of hours the assembly line
is functioning on a given day.
Find the cost of the bicycle helmets produced on a day
when the assembly line was functioning 12 hours.
(solution on next slide)
n  P  t   75t  2t 2
C  n   7n  1000
Solution to Example 5:
 Determine a function that gives the cost of producing
the helmets in terms of the number of hours the
assembly line is functioning on a given day.
 Cost  C  n   C  P  t  


 7 7
5t  2t   1000
75
 C 75t  2t 2
2
 14t 2  525t  1000
 Find the cost of the bicycle helmets produced on a day
when the assembly line was functioning 12 hours.
C  14t 2  525t  1000  $5284
Summary…
 Function arithmetic – add the functions (subtract, etc)
 Addition
 Subtraction
 Multiplication
 Division
 Function composition
 Perform function in innermost parentheses first
 Domain of “main” function must include range of “inner”
function
Class work
 Workbook Page 123-124 #13-24
Homework
 Page 114 #19-24
 Page 115 #9-16