Slide 1- 1

Sum, Difference, Product, and Quotient

Let and be two functions with intersecting domains. Then for all values

of in the intersection, the algebraic combinations of and are defined

by the following rules:

Sum: ( ) ( )

Differ

f g

x f g

f g x f x g x

ence: ( ) ( ) ( )

Product: ( )( ) ( ) ( )

( )Quotient: , provided ( ) 0

( )

In each case, the domain of the new function consists of all numbers that

belong to both the domain of and

f g x f x g x

fg x f x g x

f f xx g x

g g x

f

the domain of . g

©1999 by Design Science, Inc. 3

2

2

2 1 3

2 4x

xf g x

2

2

2

2 1

2 6 9 1

2 12 18 1

3g

x x

x

f x

x

x

Example 3• Evaluate and :

–

–

f g x g f x

3f x x

22 1g x x

22 4f g x x

22 12 17g f x x x

You can see that function composition is not commutative!

Complex fractions

Objective

• Simplify complex fractions

• Lets Review fraction rules first…………..

Simplifying Complex Fractions

A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator.

454

3xx

3x

ba9-a

ba

Example:

Adding/Subtracting Fractions

0c ,c

ba

c

b

c

a 0c ,

c

ba

c

b

c

a

712

= 512

212

+

Add . 512

212

+

Common Denominators

1. Add or subtract the numerators.2. Place the sum or difference of the

numerators found in step 1 over the common denominator.

3. Simplify the fraction if possible.

Subtract .5

6

5

7-2x

5

13-2x

5

6-7-2x

5

6

5

7-2x

Common Denominators

a.) Add .12ww

4-2w-

12ww

53w22

Example:

12ww

4-2w-53w

12ww

4-2w-

12ww

53w222

1)(w

1

1)(w

1w2

12ww

4-2w-53w2

Common Denominators

b.) Subtract

.649x

29x-x

649x

54x2

2

2

2

649x

29)x-(x-54x

649x

29x-x

649x

54x2

22

2

2

2

2

649x

24x3x

649x

29xx-54x2

2

2

22

8)(3x

3)(x

8)8)(3x(3x

8)3)(3x(x

Example:

Unlike Denominators

1. Determine the LCD.2. Rewrite each fraction as an

equivalent fraction with the LCD.3. Add or subtract the numerators

while maintaining the LCD.4. When possible, factor the

remaining numerator and simplify the fraction.

Unlike Denominators

a.)w

5

2w

3

2w

2w

w

5

w

w

2w

3

The LCD is w(w+2).

2)w(w

2)5(w

2)w(w

3w

2)w(w

105w

2)w(w

3w

2)w(w

108w

answers. acceptable also are and 2ww

108w

2)w(w

5)2(4w2

Example:

Unlike Denominators

b.)3x

1

4-4x

x The LCD is 12x(x – 1).

3x

1

1)-4(x

x

1)-4(x

1)-4(x

3x

1

3x

3x

1)-4(x

x

1)-12x(x

44x3x

1)-12x(x

1)-4(x

1)-12x(x

3x 22

This cannot be factored any further.

Example: