1. Operations With
Functions

2. Operations With Functions - Definitions
Functions that are defined by algebraic expressions can
be combined algebraically. Operations on any two
functions can be used to define new functions.
Given two functions f(x) and g(x):
The sum of f and g: (f + g)(x) = f(x) + g(x)
The difference between f and g: (f - g)(x) = f(x) - g(x)
The product of f and g: (fg)(x) = f(x) x g(x)
The quotient of f and g:
1.2

3. OR Operations With Functions - Practice
Given f(x) = 2x + 6 and g(x) = -4x - 3, write
an expression in simplest form for each:
a) (f + g)(x)
b) (fg)(x)
(f + g)(x) = f(x) + g(x)
= 2x (-4x) - 3
= -2x + 3
(fg)(x) = f(x) x g(x)
= (2x + 6)(-4x - 3)
= -8x2 - 6x - 24x - 18
= -8x2 - 30x - 18 d)
c) (f - g)(-2) OR
(f - g)(x) = f(x) - g(x)
= (2x + 6) - (- 4x - 3)
= 6x + 9
(f - g)(-2) = 6(-2) + 9 =
= -3
(f - g)(-2) = f(-2) - g(-2)
= [2(-2) + 6] - [-4(-2) - 3]
= 2 - 5
= - 3
Same
1.3

4. Operations With Functions - More Practice
Given f(x) = 3x2 + 6x and g(x) = 4x + 1, write
an expression in simplest form for each:
a) 4f(x)
b) 2g(x) - f(x)
4f(x) = 4(3x2 + 6x)
= 12x2 + 24x
2g(x) - f(x) = 2(4x + 1) - (3x2 + 6x)
= 8x x2 - 6x
= -3x2 + 2x + 2
c) (gg)(x)
(gg)(x) = g2(x)
= (4x + 1)2
= (4x + 1)(4x + 1)
= 16x2 + 8x + 1
1.4

5. Products and Quotients of Functions
To help with attendance at a local baseball park, the team owner decides to reduce ticket prices, according to the number of games that have been played. She has decided on the price function
P(x) = 35 – 0.25x, where x represents the number of games that have been played in the season.
The marketing department has investigated this plan and found that the anticipated attendance in hundreds of fans is given by N(x) = x.
a) Develop a function f(x) = P(x) N(x) and explain the meaning of this new function.
P(x) N(x) = –0.05x x + 525
b) Will the owner increase or decrease the revenue from ticket sales under the new plan? Explain.
The revenue will increase in the short term, for games up to approximately 32 or 33. Beyond that, the plan will start to decrease revenue. This is because f(x) is
a quadratic function with its maximum at
approximately 32.5.