September 10, 2014
1.5 Combinations of Functions
Objective: You will know how to find arithmetic combinations and compositions of functions.
I. Arithmetic Combinations of Functions
Let f and g be functions with overlapping domains. Then for all x common
to both domains:
1. Sum:
(f + g)(x) = f(x) + g(x)
2. Difference:
(f - g)(x) = f(x) - g(x)
3. Product
(fg)(x) = f(x) • g(x)
4. Quotient
provided g(x) = 0.
What does that mean?
Examples:
f(x) = x2 + 2x and g(x) = 2x + 1. Find the following.
1)
2)
September 10, 2014
Cont.
f(x) = x2 + 2x and g(x) = 2x + 1. Find the following.
3)
4)
Try graphing (f-g)(x) when f(x)=2x+1 and
g(x)=x2+2x-1. Then find (f-g)(2).
(To graph, enter f(x) as f1(x) and g(x) as f2(x). Define f3(x) as
f1(x)-f2(x) then graph )
,
.
Var
(Choose f1 and f2
from here.)
Answer: -2
September 10, 2014
II. Compositions of Functions
The composition of the function f with the function g is
(f o g)(x) = f(g(x)).
Read as "f of g"
The domain of (f o g) is the set of all x in the domain of g such that g(x)
is in the domain of f.
Refer to Figure 1.59 on page 53 of the text.
http://www.regentsprep.org/Regents/math/algtrig/ATP7/domaincomposite.htm
Ex: f(x) = x2 + 2x and g(x) = 2x + 1. Find the following.
1)
2)
September 10, 2014
Answer
Ex: h(x)= (x + 1)2 - x - 1. Find two functions f and g such that h(x) = (f o g)(x).
g(x) = x + 1
f(x) = x - x
2
Note: f(g(x)) is generally not the same as g(f(x)).
p.54 Exploration
Ex: The weekly cost C to manufacture a certain product is given by
C(x) = 25x + 5000. The number of units x produced in t hours is given by x(t) = 3t.
(a) Find and interpret (C o x)(t).
(C o x)(t) = 75t + 5000. This equation represents the cost after t production hours.
(b) After how many hours will the weekly cost be equal to $10,000?
10,000 = 75t + 5000
5000 = 75t
66.66 » t
After approximately 66.6 hours, the cost will equal $10,000.
Assignment:
p.58 (3-30 x3,35-37,39-41,45-47,52,57,59,63-78x3,79,82)