3.7 Composition of Functions

3.7 Composition of FunctionsFunction Frank

26515x3xf(x) = 3x

Types of OperationsAddition

Subtraction

Multiplication

Division

AdditionIf you are given two functions:

f(x) and g(x)

Their sum is:

(f + g) (x) =

f(x)+ g(x)AdditionExample:f(x) = 5x + 4 g(x) = 8x 2

(f + g) (x) = f(x) + g(x) + 8x 2 = 1 3 x + 25x + 4AdditionFind (f + g) (x) if: f(x) = 4x + 3x + 2g(x) = 2x - 5x 6

(f + g) (x) = f(x) + g(x) = 4x + 3x + 2 + 2x - 5x 6 = 6 x - 2 x - 4Now you tryFind (f + g) (x) if:f(x) = 5x + 4x 2 g(x) = 2x - 3x + 2

(f + g) (x) = f(x) + g(x) = 5x + 4x 2 + 2x - 3x + 2 = 7 x + x Types of OperationsAddition

Subtraction

Multiplication

Division

SubtractionJust as we could add to functions, we can also subtract two functions.

(f g) (x) = f(x) g(x)

SubtractionExample: Find the difference (f g) (x) if:f(x) = 8x + 4 g(x) = 7x 2

(f g) (x) = f(x) g(x) *For subtraction, use parentheses

= x + 68x + 4 7x 2( )SubtractionFind (f g) (x) if:f(x) = 8x + 4x 2 and g(x) = 3x - 2x + 1

(f g) (x) = f(x) g(x) = 8x + 4x 2 (3x - 2x + 1) = 8x + 4x 2 3x + 2x - 1 = 5 x + 6 x - 3Now you tryFind (f - g) (x) if:f(x) = 5x + 4x 2 g(x) = 2x - 3x + 2

(f - g) (x) = f(x) - g(x) = 5x + 4x 2 (2x - 3x + 2) = 5x + 4x 2 2x + 3x - 2 = 3 x + 7 x - 4

Types of OperationsAddition

Subtraction

Multiplication4. Division

MultiplicationAs with addition and subtraction, we can also multiply two functions.

For the two functions f(x) and g(x), the wefind the product by:

(f g) (x) = f(x) g(x)MultiplicationLet f(x) = 8x and g(x) = 2x

Find (f g) (x):

1) (f g) (x) = f(x) g(x)= 8x 2x= 16xMultiplication w/ Distributivef(x) = 3xg(x) = 2x 4

Find (f g) (x):

1) (f g) (x) = f(x) g(x)2) = 3x (2x 4)Here, we must use our Distributive Property Distribute the 3x over the 2x and -4= 6x- 12x

Multiplicationf(x) = 3xg(x) = x + 2h(x) = 2x - 1 Find:a. (f g) (x)b. (g h) (x)c. (f h) (x)= 3x + 6x= 2x + 3x - 2= 6x - 3xTypes of OperationsAddition

Subtraction

Multiplication4. Division

DivisionDivision of functions works just as the other 3 operations:

(f g) (x) = f(x)g(x)DivisionEx: f(x) = 2x + 3x 4 and g(x) = 5x - 2x - 1(f g) (x) = f(x)g(x)2x + 3x 4 5x - 2x - 1 =DivisionEx: f(x) = x - x 12 and g(x) = x - 2x - 8(f g) (x) = f(x)g(x)x - x 12 x - 2x - 8=(x + 3)(x - 4) (x +2 ) (x 4)=(x + 3) (x +2 )=Types of OperationsAddition

Subtraction

Multiplication4. Division

Composition of FunctionsThere is one more operation that can be performed on functions

Composition of Functions

(f g) (x) = Pronounced F of G of Xf ( g (x) )Composition of Functions f (x) = 4xand g(x) = x + 2Find (f g) (x)1. (f g) (x) = f ( )2. = f ( )x + 2This leaves us with:f (x + 2)g(x)f(x) = 4x g(x) = x + 2We need to find f(x + 2)In the f(x) function, replace the x with x + 2f(x) = 4x(x + 2)= 4x + 8Therefore, f (g (x)) = 4x + 8f(x) = 3x g(x) = 2x - 4(f g) (x) = f ( g (x) )= f (2x 4)= 3x(2x 4)= 6x - 12f(x) = x + 4 g(x) = 8x(f g) (x) = f ( g (x) )= f (8x)= + 4 x(8x)= 8x + 4f(x) = x + 5 g(x) = x - 2x(f g) (x) = f ( g (x) )= f (x - 2x)= (x - 2x) + 5= x - 2x + 5(g f) (x) = g ( f (x) )= g (x + 5)= (x + 5) - 2(x+5)= (x + 5) - 2x 10f(x) = 2x ; g(x) = x + 2(f g) (x) = f ( g (x) )= f (x + 2)= 2 (x + 2)= 2x + 4(g f) (x) = g ( f (x) )= g (2x)= 2x + 2f(x) = 2x ; g(x) = x - 5Find (f g) (x) = f (g (x) )= f (x 5)= 2 (x 5) Find (g f ) (x) = g ( f (x) )= g (2x)= 2x - 5Classwork: Find (f g) (x)f(x) = x ; g(x) = x + 4

f(x) = 4x ; g(x) = x 3

f(x) = 2x ; g(x) = x + 2

f(x) = x ; g(x) = 2x

f(x) = x + 1 ; g(x) = 3x= x + 4= 4x - 12= 2x + 4= 4x= 3x + 1Substituting with NumbersWe can use these same formulas to calculate problems substituting numbers in for x.For example: Find (f + g) (3) for the two functions: f(x) = 5x + 2 and g(x) = 2x - 2= f(3) + g(3)= 5(3) + 2+ 2(3) - 2=15 + 2 + 6 - 2= 21f(x) = 8x + 2 g(x) = 4xFind: a. (f g) (4)b. (f g) (1)c. (f g) (2)= f(4) g(4)= 8 (4) + 2- 4(4)= 32 + 2 - 16= 18= 18= f(1) g(1)= (8(1) + 2) 4(1)= (10) (4)= 40= 40= f(2)g(2)=188=94