1.3 Creating New Polynomial Functions: An Introduction to ...

Adding, Subtracting, and Multiplying
Polynomial Functions
For the polynomial functions f and g,
• the sum is ( f g)(x) f (x) g(x)
• the difference is ( f g)(x) f (x) g(x)
• the product is ( fg)(x) f (x) g(x)
The domain of f ± g and fg is the set of all x, for which both f and g are
defined. Any x belonging to the domain of f and to the domain of g belongs
to the domain of f (x) ± g(x) or f (x) g(x).
EXAMINING THE CONCEPT
input → output
x → g(x) → f (g(x))
g f
input → output
The Composition of Functions
What happens when the output of one function is the input of another
You can think of a function as an input-output machine. The input set
is the domain of the function, and the output set is the range of the function.
In the first step of the diagram, the x is an input and g(x) is an output.
In the second step, g(x) is an input and f (g(x)) is an output.
Combining two functions in this way is called composition. Composition is
the result of substituting the output of one function into another. The new
function is called the composite of f and g and is written f g.
Example 3
Determining the Equation of a Composite Function
Recall the situation in Setting the Stage. Rhona earns a daily wage according to
R(h) 28h 22. The job site where Rhona works requires all employees to
become members of a union. Rhona then pays 1.5% of her daily earnings as
union dues. Determine the function that represents her daily union dues.
Hours Worked
h
1
2
3
.
.
.
.
8
R
Daily Earnings
R(h)
50
78
106
.
.
.
.
246
Solution
U
Union Dues
U(R(h))
0.75
1.17
1.59
.
.
.
.
3.69
This situation involves two functions,
Daily Earnings: R(h) 28h 22
Daily Union Dues: U(R(h)) 0.015 R(h)
Examine the mapping that shows how the number of
hours worked per day is related to the daily union dues.
The domain for Rhona’s daily earnings is the number of
hours worked. But the domain for the daily union dues
are her daily earnings, not the number of hours worked.
In other words, the domain of the substitution function is
the range of the substituted function.
1.3 CREATING NEW POLYNOMIAL FUNCTIONS: AN INTRODUCTION TO COMPOSITION 31