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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
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