Operations with Functions

Objective: Perform operations with functions and find the composition of

two functions

32 x 10x 62 xx

x5 xx 39 2

Operations with Functions

• For all functions f and g:

• Sum: (f + g)(x) = f(x) + g(x)• Difference (f – g)(x) = f(x) – g(x)• Product (fg)(x) = f(x)g(x)• Quotient (f/g)(x) = f(x)/g(x)

Example 1

Example 1

Example 1

Try This

• Let and .• Find: (f + g)(x) (f – g)(x)

5.2127)( 2 xxxf 57)( 2 xxg

Try This

• Let and .• Find: (f + g)(x) (f – g)(x)

5.2127)( 2 xxxf 57)( 2 xxg

)57()5.2127())(( 22 xxxxgf

5.212 x

Try This

• Let and .• Find: (f + g)(x) (f – g)(x)

5.2127)( 2 xxxf 57)( 2 xxg

)57()5.2127())(( 22 xxxxgf

5.212 x

)57()5.2127())(( 22 xxxxgf

5.71214 2 xx

Example 2

Example 2

Example 2

Try This

• Let and .• Find: (f g)(x) (f/g)(x)

13)( 2 xxf 2)( xxg

Try This

• Let and .• Find: (f g)(x) (f/g)(x)

13)( 2 xxf 2)( xxg

263)2)(13())(( 232 xxxxxxfg

Try This

• Let and .• Find: (f g)(x) (f/g)(x)

13)( 2 xxf 2)( xxg

263)2)(13())(( 232 xxxxxxfg

2;2

13))(/(

2

xx

xxgf

Example 3

Example 3

Example 3

Try This

• Let and .• Find: and

32)( 2 xxf xxg 2)(

gf fg

Try This

• Let and .• Find: and

32)( 2 xxf xxg 2)(

gf fg

383)2(2)2( 22 xxxfgf

Try This

• Let and .• Find: and

32)( 2 xxf xxg 2)(

gf fg

383)2(2)2( 22 xxxfgf

64)32(2)32( 222 xxxgfg

Example 4

Example 4

Example 4

Example 5

• Let and . • Find

43)( xxf 2)( xxg

)3)(( gf

Example 5

• Let and . • Find• You have two choices. • First, let’s find f(g(x)) and evaluate it at x = 3.

43)( xxf 2)( xxg

)3)(( gf

Example 5

• Let and . • Find• You have two choices. • First, let’s find f(g(x)) and evaluate it at x = 3.

43)( xxf 2)( xxg

)3)(( gf

43))(( 2 xxgf

314)3(3))3(( 2 gf

Example 5

• Let and . • Find• You have two choices. • Second, we can find g(3) and put that answer into f.

43)( xxf 2)( xxg

)3)(( gf

Example 5

• Let and . • Find• You have two choices. • Second, we can find g(3) and put that answer into f.

43)( xxf 2)( xxg

)3)(( gf

9)3( g

314)9(3)9( f

Homework

• Page 115• 11-55 odd