1. 1.7 Composition of Functions

2. Composition of Functions
Yesterday we went over combining functions using:
Addition, Subtraction, Multiplication, Division
The last type of combination is Composition of Functions

3. Composition of Functions

4. Composition of Functions
Given two functions f and g, the function
(f ○ g) x = f [ g (x)] and
(g ○ f) x = g [ f (x)]

5. f(x) = 3x² g(x) = 2x
(f ○ g) (x)
(g ○ f) (x)
(g ○ g) (x)

6. f(x) = 3x² g(x) = 2x = f [g(x)] = 3 (2x)² = 3 (4)(x²) = 12x²

7. f(x) = 3x² g(x) = 2x
= g [f(x)]
(g ○ f) (x)
= g(3x²)
= 2(3x²)
= 6x²

8. f(x) = 3x² g(x) = 2x
= g [g(x)]
(g ○ g) (x)
= g(2x)
= 2(2x)
= 4x

9. f(x) = x g(x) = 4 - x²
(f ○ g) (x)
(g ○ f) (x)
(f ○ f) (x)

10. f(x) = x + 2 g(x) = 4 - x² = f [g(x)] = (4 - x²) = 6 - x² + 2

11. f(x) = x + 2 g(x) = 4 - x² = g [f(x)] = 4 - (x + 2)²

12. f(x) = x + 2 g(x) = 4 - x² = f [f(x)] = (x + 2) + 2 = x + 4

13. 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

14. 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

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

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