Understanding Composite Functions in Mathematics

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Composite functions in mathematics involve applying two functions in succession, yielding a new function known as the composite. By evaluating functions in a specific order and considering their ranges and domains, composite functions provide a powerful tool for mathematical analysis.

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10.3 Composite Functions = = + = + ( ) f x 4 ( ) g x 6 x x ( ) h x 5 x = ( ( )) f g x x ( x ) 6 4 6 x+ + = = = x ( ( ) g x ) 4 4 f How would you define composite functions? ( ) + 4 24 x Math30-1 1

Composition of Two Functions When two functions are applied in succession, the resulting function is called the composite of the two given functions. (g o f)(x) = g(f(x)) The range of f(x) becomes the domain of g(x). Evaluate f(x) then use the answer as the input for g(x). Notice: The function is evaluated from the inside to the outside. x is the input into the f function to get y y is the input into the g function to get z Math30-1 2

10.3 Function Composition 10.3 Function Composition = ( ) f x y ( ) = ( ) y f g x = ( ) g x y f ( ) = = ( ) y g f x ( ) f x y g Math30-1 3

= = + = + ( ) f x 4 ( ) g x 6 x x ( ) h x 5 x = ( ( )) h f x (4 ) h x ( ) = + 4 5 x 5 4 = + 4 5 x x , Math30-1 4

Evaluating a Composition of a Functions Given h(x) = 4x + 3, determine the following: a) (h h)(x) b) (h h)(-3) (h h)(x) = h(h(x)) = h(4(x) + 3) = h(4x + 3) (h h)(-3) = h(h(-3)) = h(4(-3) + 3) = h(-9) h(4x + 3) = 4(4x + 3) + 3 = 16x + 12 + 3 = 16x + 15 h(-9) = 4(-9) + 3 = -33 Note: (h h)(x) (hh)(x) (hh)(x) = h(x) xh(x) Math30-1 5

= = + ( ) h x 2 5 x 2 = ( ) g x 3 x ( ) f x 1 x Given Determine an expression in simplest form for ( 2 1 5 x ) = x ( ( )) h f x , 1 Why are there restrictions? ( ) = + ( )( ) ( ) k x 2 2 5 x = ( ) k x h g ( ( h f x ) ) ( = ( ) k x 1 ) h g ( x = ( ) k x 2 1 x x , 1 ) 2 = + ( ) k x 1 3 x ( ) = + ( ) k x 2 h x Math30-1 6

+ 2 ( ) 4 2 x x ( ) 4 f f = ( ) f x Using the function determine the value of f + 2 4 4 2 4 ( ) ( ) 4 = f f ( ) ( ) 4 ( ) = 10 f f f + 2 10 10 2 4 ( ) ( ) 4 = f f ( ) ( ) 4 = 13 f f Math30-1 7

= ( ) g x = ( ) f x log x 5x 5 Determine the expressions for h(x)= f(g(x)) and k(x) = g(f(x)) = = ( ) h x ( ( )) f g x ( ) k x ( ( )) g f x ( ) 5x ( ) h x = log ( ) k x = log x 5 5 5 = ( ) h x x = ( ) k x x The situation when f(g(x)) = g(f(x)) = x indicates that the original functions are inverses. Math30-1 8

Use the graph to determine the value of ( )(3) = = = + + (3) + (3) f g f g = ( ) f x y 0 ( 12) 12 ( ) ( ) 2 ( ) 4 = = g ( ) g x y g f = 12 Math30-1 9