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7.4 Composition of Functions

7.4 Composition of Functions. 2/26/2014. Function:. A function is a special relationship between values: Each of its input values gives back exactly one output value . It is often written as "f(x)" where x is the input value. Ex 1: G find:. a.) Solution: Substitute 0 for x in 3x+ 1

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7.4 Composition of Functions

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  1. 7.4 Composition of Functions 2/26/2014

  2. Function: A function is a special relationship between values: Each of its input values gives back exactly one output value. It is often written as "f(x)" where x is the input value.

  3. Ex 1: Gfind: a.) Solution: Substitute 0 for x in 3x+ 1 3(0) + 1 = 1 = 1 b.) 3(½) + 1 = 1 + 1 = 2 c.) 3(a) + 1 = 3a + 1 d.) 3(a+3) + 1 = 3a + 9 +1 = 3a+10 e.) = 3a + 4

  4. Ex 2: Gfind: a.) = -5 b.) = - c.) = d.) = e.) 3 =

  5. y 8 g 6 f 4 2 x –8 –6 2 4 6 8 –4 –2 –2 –4 Use the graph to find: f(0) Solution: at x = 0, the graph of the f (red) function is at 2. f(0) = 2 b. g(0) Solution: at x = 0, the graph of the g (black) function is at 3 g(0) = 3

  6. Composition of functions occurs when you insert one function into another. In effect, the range (output) of the inside function becomes the domain (input) of the outside function. The notation for composition of functions is either Or Note: which means order matters!

  7. y 8 g 6 f 4 2 x –8 –6 2 4 6 8 –4 –2 –2 –4 Use the graph to find: c.) f(g(0)) Solution: Evaluate g(0) first, and from the previous problem (b), g(0) = 3. Then look at the graph of the f function and see what the y component when x = 3. At x = 3 the red graph is at -1 f(g(0)) = -1

  8. y 8 g 6 f 4 2 x –8 –6 2 4 6 8 –4 –2 –2 –4 Use the graph to find: d.) g(f(0)) Solution: Evaluate f(0) which in problem (a) is 2. Then look at the graph of the g function and see what the y component when x = 2. At x = 2 the black graph is at 5. g(f(0)) = 5

  9. y 8 g 6 f 4 2 x –8 –6 2 4 6 8 –4 –2 –2 –4 Use the graph to find: a.) f(g(-1)) and b.) g (f(-1)) a.) Solution: Evaluate g(-1) g(-1) = 2 then evaluate f(2) f(2) = 0 b.) Solution: Evaluate f(-1) f(-1) = 3 then evaluate g(3) g(3) = 6

  10. Given: Evaluate the following expressions: f(-1) g(-1) f(g(-1)) g(f(-1)) a.) b.) c.) d.)

  11. Homework: WS 7.4 #1-9

  12. Let and . Find: ( ) ( ) g x x 1 4x2 + = f x = SOLUTION a. b. ( ) ( ) ( ) ( ) ( ) – h x f x + g x f x g x = ( ) ( ) – x 1 x 1 + + 4x2 4x2 + = – – 4x2 4x2 x 1 x 1 + + = In both parts and , the domains of f and g are all real numbers. So, the domain of h is all real numbers. ( ) ( ) a b Example 1 Add and Subtract Functions a. b. ( ) ( ) – ( ) ( ) f x g x f x + g x

  13. 1 ( ) ( ) f x x3 g x 2x = = x2 = 2 2x4 = Example 2 Multiply and Divide Functions Let and . Find: ( ) f x a. b. ( ) ( ) f x g x • ( ) g x

  14. ANSWER – 4x 1 3x , x 1 = – x 1 2. f ( ) ( ) x g x • – 3x2 3x, Perform Function Operations Checkpoint Let and . Find ( ) ( ) – f x 3x g x x 1 = = 1. ( ) ( ) f x + g x ( ) f x 3. ( ) g x

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