Transcript Composite Functions - Foothill Technology High School

Composite Functions
What Are They?
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.
Example 1
Given
f x   x 3 g x   x  2
the composite function
f  gx  f gx  f x  2  x  2  x3  6x2  8x  8
3
Replace g(x) with x+2
Replace the variable
x in the f function
with x+2
Expand
Example 2
2
Given hx  2x  4 k x  x  2 the composite function
k h  x   k  h  x   k

 
2x  4 
The result of
the function
h becomes
the input to
k
2x  4

2
 2   2x  4  2  2x  2
Replace the
variable x in
k(x) with 2 x  4
Simplify
Example 2 Con’t.
Now see what happens when we take the same two functions and
reverse the order of the composition.
hx  2x  4 k x  x2  2

The composite function



h k  x   h  k  x   h x2  2  2 x2  2  4  2x2  4  4  2x2  2 x
Notice, the result here is not the same as the previous result. This is
usually the case with composite functions. Changing the order of
the composition (changing which function is the “inner” function and
which is the “outer” function) usually changes the result.
Problem 1
For the functions f  x  
f g  x
1
g  x   3x  5 find
x
(click mouse to see answer)
1
f  g  x    f  3x  5 
3x  5
g f  x
(click mouse to see answer)
3
1
1
g  f  x   g    3   5   5
x
 x
 x