with function operations addition: subtraction: multiplication: division:
TRANSCRIPT
with
Function Operations
• Addition:
• Subtraction:
• Multiplication:
• Division:
)()())(( xgxfxgf
)()())(( xgxfxgf
)()())(( xgxfxgf
0)(,)(
)()(
xg
xg
xfx
g
f
Examples:If and 1) Find (f + g)(x)
2) Find (f - g)(x)
3) Find
4) Find
))(( xgf
132)( 2 xxxf xxg 5)(
)(xg
f
Examples:If and 132)( 2 xxxf xxg 5)(
1) Find (f + g)(3)
2) Find (f - g)(0)
3) Find
4) Find
)2)(( gf
)3(
g
f
OF
Definition of composite functions
Suppose f and g are functions such that the range (output) of g is the subset of the domain (input) of f. Then the composite function
f g
can be described by the equation
f g (x) f (g(x)).
Let’s do an example together.
EX: If f (x) x2 3x 1 and g(x) 2x 3,
Find ( f g)(x) and (g f )(x).
( f g)(x) f (2x 3)
f (2x 3) (2x 3)2 3(2x 3) 1
• Substitute 2x-3 in for g(x):
• Substitute into f(x):
Next
( f g)(x) 4x2 18x 19
Simplify:
f (2x 3) (2x 3)2 3(2x 3) 1
4x2 12x 9 6x 9 1
4x2 18x 19
Now lets find (g f )(x)
(g f )(x) g(x2 3x 1)
g(x 2 3x 1) 2(x2 3x 1) 3• Substitute f(x) into g(x):
(g f )(x) 2x2 6x 2 3
2x2 6x 1
• Substitute for g(x):x2 3x 1
• Simplify:
Another example
If f (x) x 2 and g(x) x2 3, find f (g(x))
and g( f (x)).
f (g(x)) f (x2 3)
x2 3 2
x2 1
Next
g( f (x)) g(x 2)
(x 2)2 3
x2 4x 1
Try these on your own:Find f (g(x)) and g( f(x)).
a) f (x) 2x 7 ; g(x) 3x 1
b) f (x) x2 2x ; g(x) x 3
Answers:
a) ( f g)(x) 6x 2
b) (g f )(x) x2 2x 3
Now we will go over how to find a value of composite of functions.
If f (x) x2 2x 1 ; g(x) 2x 3 ; h(x) 4x
Find each value.
a) ( f g)( 2) f ( 2x 3) Substitute in g(x).
f ( 2( 2) 3) Substitute into g(x).
f (7) SimplifyNext
f (7) 72 2(7) 1 Substitute into f(x).
( f g)( 2) 64 Simplify
b) (g f )( 2) g(( 2)2 2( 2) 1)
g(1)
2(1) 3
1
Now you find the values using the same directions as in the last examples.
1) (h f )(3)
2)(g h)( 1)
Answer: 64
Answer: -5