C. K.
Lee
2. Operations on Functions and Graphs
2.1 New Functions from Old
A. Operations of functions
• If f and g are functions:
(1) sum of f and g → (f + g)(x) = f(x) + g(x)
(2) difference of f and g → (f − g)(x) = f(x) − g(x)
(3) product of f and g → (f g)(x) = f(x)g(x)
f f(x)
(4) quotient of f and g → ( ) (x) = , g(x) ≠ 0,
g g(x)
where Df+g = Df-g = Dfg = Df∩Dg and Df/g = (Df∩Dg)\{g(x)=0}
• Ex 1) For f(x) = x 2 + 1 and g(x) = 3x + 5, find:
(a) (f+g)(1)
(b) (f-g)(-3)
(c) (f g)(5)
f
(d) (g) (0)
• Ex 2) For f(x) = 3x 2 − 2x and g(x) = x 2 − 2x − 3, find each operation and give the domain
of the new function.
(a) f − g
f
(b) 3g
M151_2.1-1
C. K. Lee
B. Composite Functions
Composition of functions means that one function is applied after another.
Definition:
Given two functions f and g, the composite function, denoted by f g , is defined by
( f g )(x ) = f (g (x )), where 𝐷𝑓𝑜𝑔 = {𝑎𝑙𝑙 𝑥 ∈ 𝐷𝑔 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑔(𝑥) ∈ 𝐷𝑓 } ⫅ 𝐷𝑔.
Ex 1) Suppose that f(𝑥 ) = 2𝑥 2 − 3 and g(𝑥 ) = 4𝑥. Evaluate each composite function.
(1) ( f g )(x)
(2) (g f )(x )
1 4
Ex 4) Suppose that f(𝑥 ) = 𝑥+2 and g(𝑥 ) = 𝑥−1. Find:
(1) f g
(2) D f g
(3) g f
(4) D g f
(5) f f
(6) D f f
Ex 5) Suppose 𝑓 (2) = 3, 𝑓 (3) = 4, 𝑓 (5) = 7, 𝑔(2) = 5, and 𝑔 (3) = 8. In each part find the
indicated function value.
𝑓(𝑥)
(a) If ℎ(𝑥 ) = 3𝑓(𝑥 ) + , find ℎ(2).
𝑔(𝑥)
(b) If 𝑘 = 𝑓 ∘ 𝑔 − 𝑓𝑔, find 𝑘(2).
M151_2.1-2
C. K. Lee
C. Decomposing a function as a composition
Function decomposition identifies how simple parts make up more complicated functions.
Ex A) Find f(x) and g(x) such that h( x) = ( f g )( x) . Answers may vary.
(1) h(x) = (2𝑥 − 3)5
1
(2) h(𝑥) = (𝑥+3)2
x−5
(3) h(x ) =
x+2
Ex 7) A rock tossed in a pond makes an expanding circle of water ripples. The area of that circle at
time t seconds is 𝐴(𝑡) = 𝜋(1 + 2𝑡)2 square inches. Find a function 𝑓 so that 𝐴 = 𝑓 ∘ 𝑟, where
𝑟(𝑡) = 1 + 2𝑡
M151_2.1-3
