6.3 Perform Function Operations and Composition

Power function

A function of the form y= ax^b where a is a real number and b is a rational number

Composition

The composition of a function g with a function f is h(x) = g(f(x)). The domain of h is the set of all x-values such that x is in the domain of f and f(x) is in the domain of g.

OPERATIONS ON FUNCTIONS

Let f and g be any two functions. A new function h can be defined by performing any of the four basic operations on f and g.

Operation and Definition	Example: f(x) = 3x, g(x) = x + 3
Addition
h(x) = f(x) + g(x)	h(x) = 3x + (x + 3)
	= 4x +3
Subtraction
h(x) = f(x) - g(x)	h(x) = 3x - (x + 3)
	= 2x - 3
Multiplication
h(x) = f(x) · g(x)	h(x) = 3x(x + 3)
	= 3x^2 + 9x
Division
h(x) =	h(x) =

The domain of h consists of the x-values that are in the domains of both f and g . Additionally, the domain of a quotient does not include x-values for which g(x) = 0.

Example 1

Let f(x) = 3x^1/2 and g(x) = -5x^1/2. Find the following.

a. f(x) + g(x)

b.f(x) - g(x)

c. the domains of f + g and f - g

Solution

a. f(x) + g(x) = 3x^1/2 + (-5x^1/2)

= [3 + (-5)] x^1/2 = -2x^1/2

b. f(x) - g(x) = 3x^1/2 - (-5x^1/2)

= [3 - (-5)] x^1/2 = 8x^1/2 The functions f and g each have the same domain: all nonnegative real numbers . So, the domains of f + g and f - g also consist of all nonnegative real numbers .

Example 2

Let f(x) = 7x and g(x) = x^1/6. Find the following.

a. .f(x) · g(x)

b.

c. the domains of f · g and

Solution

a.

x

7

f(x) · g(x) = 7(x)(x^{1/ 6}) = 7x^{(1 + 1/ 6)} = 7x^{7/ 6}

b.

-

=

=

x

x

x

7

7

1

6

(

1

1

6

)

5

6

=

c. The domain of f consists of all real numbers, and the domain of g consists of all nonnegative real numbers. So, the domain of f · g consists of all nonnegative real

numbers. Because g(0) = 0 , the domain of is restricted to all positive real numbers. .

COMPOSITION OF FUNCTIONS

The composition of a function g with a function f is h(x) = g(f(x)) . The domain of h is the set of all x-values such that x is in the domain of f and f(x) is in the domain of g .

Example 3

Let f(x) = 6x^-1 and g(x) = 3x + 5. Find the following.

a. f(g(x))

b. g(f(x))

c. f(f(x))

d. the domain of each composition

x

-

+

=

1

6

(

3

5

)

6

Solution

a.

x

+

3

5

f(g(x)) = f(3x +5) =

b.

18

g(f(x)) = g(6x^-1)

x

x

x

-

-

+

=

+

=

+

1

1

3

(

6

)

5

18

5

5

=

c. f(f(x)) = f(6x^-1) = 6(6x^-1)^-1.= 6(6^-1x) = 6⁰x = x

d. The domain of f(g(x)) consists of all real numbers except x = because

G()=0 is not in the __domain of f__. (Note that f(0) = , which is _undefined__.) The domains of g(f(x)) and f(f(x)) consist of __all real numbers__ except x = __0__, again because __0 is not in the domain of f__.