Algebra of Functions; Composite Functions

Let f and g be two functions. The sum, f + g, the difference, f – g, the product, fg, and the quotient, are functions defined as follows:
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:
The domains of the sum, difference, and product functions are the set of all real numbers common to the domains of both f and g. The domain of includes all real numbers common to the domains of f and g for which g(x) ¹ 0.
Example. Let f(x) = 4x - 1 and g(x) = (2x + 1). Find: a) (f + g)(5), b) (f – g)(2),
c) (fg)(-1), and d) .
Solution. a) f(5) = 4(5) - 1 = 4(125) – 1 = 500 – 1 = 499
g(5) = (2(5) +1)= (10 + 1)= 11= 121
(f + g)(5) = f(5) + g(5) = 499 + 121 = 620
b) f(2) = 4(2) - 1 = 4(8) – 1 = 32 – 1 = 31
g(2) = (2(2) +1)= (4 + 1)= 5= 25
(f – g)(2) = f(2) – g(2) = 31 – 25 = 6
c) ) f(-1) = 4(-1) - 1 = 4(-1) – 1 = -4 – 1 = -5
g(-1) = (2(-1) +1)= (-2 + 1)= (-1)= 1
(fg)(-1) = f(-1) × g(-1) = -5(1) = -5
d) f(0) = 4(0) - 1 = 4(0) – 1 = 0 – 1 = -1
g(0) = (2(0) +1)= (0 + 1)= 1= 1
=
Example. Given f(x) = and g(x) = x - 5x , find f + g, f – g, fg, and and their domains.
Solution. (f + g)(x) = f(x) + g(x) = + x - 5x
(f - g) (x) = f(x) – g(x) = - (x - 5x) = - x + 5x
(fg)(x) = f(x) × g(x) = (x - 5x)
= x
The domain of f consists of values for which the radicand, x – 1, is nonnegative; that is
x – 1 > 0. Solving this inequality gives us x > 1. So, the domain of f is [1,¥ ).
The domain of g is the set of all real numbers because any real number substituted for x in the expression x - 5 yields a result that is a real number.
Numbers common to the domains of both f and g are in the interval [1, ¥ ). So the domain of f + g, f – g, and fg is [1, ¥ ).
To find the domain of , we must find values for which g(x) = 0:

x - 5x = 0	Set g(x) equal to 0
x(x – 5) = 0	Solve by factoring
x = 0 or x – 5 = 0 x = 5

0 and 5 must be excluded from the domain. Numbers in the domain are in the interval [1,¥ ), but not including 0 or 5. We can write this set of numbers in the following way:
[1,5) È (5, ¥ ).

Composition of Functions
Given two functions f and g, the composition of f and g, denoted by , is defined by ()(x) = f[g(x)] and the composition of g and f, denoted by , is defined by ()(x) = g[f(x)].
Example. Let f(x) = and g(x) = 2x – 1 . Find a) ()(x) , b) ()(x),
c) ()(5), and d) ()(16).
Solution.

a) ()(x) = f[g(x)]	Definition of
= f[2x – 1]	Replace g(x) with 2x – 1
=	Use f(x), and replace x with 2x – 1

b) ()(x) = g[f(x)]	Definition of
= g[]	Replace f(x) with
= 2 - 1	Use g(x), and replace x with

c) ()(5) = f[g(5)]	Definition of
= f[9]	Find g(5) by substitution: g(5) = 2(5) – 1 = 10 – 1 – 9
= 3	Find f(9) by substitution: f(9) = = 3

d) ()(16) = g[f(16)].	Definition of
= g[4]	Find f(16) by substitution: f(16) = = 4
= 7	Find g(4) by substitution: g(4) = 2(4) – 1 = 8 – 1 - 7

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