Section 1.4 – Building Functions from Functions
Objectives:
- Build new functions by adding, subtracting, multiplying, dividing, and composing functions.
- Determine the domain and range of new functions.
- Determine input and output values of a new function based on the building functions values given
algebraically, graphically, and numerically.
- Identify two possible functions that could be composed to make a new function (called decomposing a
function).
Given below are the functions to be used in order to practice building functions.
f ( x)  3 x  2
g ( x)  x 2  1
h( x)  x  4
Write the new function and determine its domain.
Function
1. ( f  g )( x)
2. ( fg )( x)
3. ( g  h)( x)
h
4. ( )( x)
f
Domain
j ( x)  2 x  5
Notice that the circle is open and this is how you know to compose rather than multiply.
Again determine the new function and its domain/range.
5. ( f g )( x)
Notice that the input values (domain) for f(x) come from the output (range) of g(x).
f(x)
g(x)
f(g(2))
X
3
2
2
Y
11
3
?
Determine the new function and its domain.
Function
Domain
6. ( h g )( x)
7. ( g h)( x)
8. f ( j ( x ))
Use the newly defined functions through their graphs to find the
new function values.
9. g ( f (5))  ________ (approximate if needed )
10. g ( f ( x))  2. what is the value of x ? __________
11. What is the domain of g ( f ( x)) ?
Use the table below to answer the following questions.
X
-2
-1
0
2
3
F(x)
4
5
3
1
2
G(x)
3
2
0
-1
5
H(x)
2
1
-1
5
0
12. ( f g )(2)  _______
13. ( f  h)(2)  __________
g
14. ( )(3)  _______
f
15. (h f )(_____)  0
One of the f functions in Column B can be composed with one of the g function in Column C to yield each
of the basic f g functions in column A. Justify why your composition of functions works.
f g
Column A
x
F
Column B
x
x
2x  3
x3
x5
G
Column C
x 0.6
x2
x3
2
Decomposing functions is when we reverse the process and find functions that would composite to the
function given.
Example: h( x)  ( x  1)2  3( x  1)  4 and was build so h( x)  f ( g ( x)) . What are two functions for f and g
that would work?
f ( x)  x 2  3x  4 and g ( x)  x  1
Decompose the following function.
16. f ( g ( x))  h( x) and
h( x)  x3  1
Do you think that everyone came up with the same functions for f and g? Why or why not?