1.7 Composition of Functions

Composition of Functions

• Yesterday we went over combining functions using:– Addition, Subtraction, Multiplication, Division

• The last type of combination is Composition of Functions

Composition of Functions

1 3x g(x) x f(x) 2

g(-2) isWhat f(-5) findThen

25 (-2)] [g f Therefore,

Composition of Functions

• Given two functions f and g, the function (f ○ g) x = f [ g (x)] and

(g ○ f) x = g [ f (x)]

f(x) = 3x² g(x) = 2x

a) (f ○ g) (x)

b) (g ○ f) (x)

c) (g ○ g) (x)

f(x) = 3x² g(x) = 2x

a) (f ○ g) (x)

= f(2x)= 3 (2x)²= 3 (4)(x²)= 12x²

= f [g(x)]

f(x) = 3x² g(x) = 2x

a) (g ○ f) (x)

= g(3x²)= 2(3x²)= 6x²

= g [f(x)]

f(x) = 3x² g(x) = 2x

a) (g ○ g) (x)

= g(2x)= 2(2x)= 4x

= g [g(x)]

f(x) = x + 2 g(x) = 4 - x²

a) (f ○ g) (x)

b) (g ○ f) (x)

c) (f ○ f) (x)

a) (f ○ g) (x)

= f(4 - x²)= (4 - x²)= 6 - x²

= f [g(x)]

f(x) = x + 2 g(x) = 4 - x²

+ 2

a) (g ○ f) (x)

= g(x + 2)= 4 - (x + 2)²= 4 – (x² + 4x + 4)= -x² - 4x

= g [f(x)]

f(x) = x + 2 g(x) = 4 - x²

a) (f ○ f) (x)

= f(x + 2)= (x + 2) + 2= x + 4

= f [f(x)]

f(x) = x + 2 g(x) = 4 - x²

Components of Compositions

• When given a composite function, you may also be asked to find its components

i.e. the two functions f(x) and g(x)

There will be more than one answer for these types of problems

Suppose h(x) = (3x – 5)³

Find f(x) and g(x) such that (f○g)(x) = h(x)

Look for an “inner” function and an “outer” function

Suppose h(x) =

Find f(x) and g(x) such that (f○g)(x) = h(x)

13

1

x

Suppose h(x) =

Find f(x) and g(x) such that (f○g)(x) = h(x)

4)32( x