3. 2 6 5 15 x 3x f(x) = 3x

4. 1. Addition 2. Subtraction 3. Multiplication 4. Division

5. If you are given two functions: f(x) and g(x) Their sum is: (f + g) (x) = f(x) + g(x)

6. EExample: f(x) = 5x + 4 g(x) = 8x – 2 (f + g) (x) = f(x) + g(x) + 8x – 2 = 1 3 x + 2 5x + 4

7. FFind (f + g) (x) if: f(x) = 4x² + 3x + 2g(x) = 2x² - 5x – 6 1. (f + g) (x) = f(x) + g(x) 2. = 4x² + 3x + 2 + 2x² - 5x – 6 3. = 6 x ² - 2 x - 4

8. FFind (f + g) (x) if: f(x) = 5x² + 4x – 2 g(x) = 2x² - 3x + 2 1. (f + g) (x) = f(x) + g(x) 2. = 5x² + 4x – 2 + 2x² - 3x + 2 3. = 7 x ² + x

9. 1. Addition 2. Subtraction 3. Multiplication 4. Division

10.  Just as we could add to functions, we can also subtract two functions.  (f – g) (x) = f(x) – g(x)

11. EExample: Find the difference (f – g) (x) if: f(x) = 8x + 4 g(x) = 7x – 2 1. (f – g) (x) = f(x) – g(x) *For subtraction, use parentheses = x + 6 8x + 4 –7x – 2 ( )

12. FFind (f – g) (x) if: f(x) = 8x² + 4x – 2 and g(x) = 3x² - 2x + 1 1. (f – g) (x) = f(x) – g(x) 2. = 8x² + 4x – 2 – (3x² - 2x + 1) 3. = 8x² + 4x – 2 – 3x² + 2x - 1 4. = 5 x ² + 6 x - 3

13. FFind (f - g) (x) if: f(x) = 5x² + 4x – 2 g(x) = 2x² - 3x + 2 1. (f - g) (x) = f(x) - g(x) 2. = 5x² + 4x – 2 – (2x² - 3x + 2) 3. = 5x² + 4x – 2 – 2x² + 3x - 2 4. = 3 x ² + 7 x - 4

14. 1. Addition 2. Subtraction 3. Multiplication 4. Division

15.  As with addition and subtraction, we can also multiply two functions. For the two functions f(x) and g(x), the we find the product by: (f g) (x) = f(x) g(x)

16.  Let f(x) = 8x and g(x) = 2x² Find (f g) (x): 1) (f g) (x) = f(x) g(x) = 8x 2x² = 16x³

17. f(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 = 6x²- 12x

19. f(x) = 3xg(x) = x + 2h(x) = 2x - 1 Find: a. (f g) (x) b. (g h) (x) c. (f h) (x) = 3x² + 6x = 2x² + 3x - 2 = 6x² - 3x

20. 1. Addition 2. Subtraction 3. Multiplication 4. Division

21.  Division of functions works just as the other 3 operations: (f ÷ g) (x) = f(x) g(x)

22.  Ex: f(x) = 2x² + 3x – 4 and g(x) = 5x² - 2x - 1 (f ÷ g) (x) = f(x) g(x) 2x² + 3x – 4 5x² - 2x - 1 =

23.  Ex: f(x) = x² - x – 12 and g(x) = x² - 2x - 8 (f ÷ g) (x) = f(x) g(x) x² - x – 12 x² - 2x - 8 = (x + 3)(x - 4) (x +2 ) (x – 4) = (x + 3) (x +2 ) =

24. 1. Addition 2. Subtraction 3. Multiplication 4. Division

25.  There is one more operation that can be performed on functions  Composition of Functions (f ○ g) (x) = Pronounced “F of G of X” f ( g (x) )

26. f (x) = 4xand g(x) = x + 2 Find (f ○ g) (x) 1. (f ○ g) (x) = f ( ) 2. = f ( ) x + 2 This leaves us with:f (x + 2) g(x)

27. We need to find f(x + 2) In the f(x) function, replace the x with x + 2 f(x) = 4x (x + 2) = 4x + 8 Therefore, f (g (x)) = 4x + 8

28. (f ○ g) (x) = f ( g (x) ) = f (2x – 4) = 3x(2x – 4) = 6x - 12

29. (f ○ g) (x) = f ( g (x) ) = f (8x) = + 4 x(8x) = 8x + 4

30. (f ○ g) (x) = f ( g (x) ) = f (x² - 2x) = (x² - 2x) + 5 = x² - 2x + 5 (g ○ f) (x) = g ( f (x) ) = g (x + 5) = (x + 5)² - 2(x+5) = (x + 5)² - 2x – 10

31. (f ○ g) (x) = f ( g (x) ) = f (x + 2) = 2 (x + 2) = 2x + 4 (g ○ f) (x) = g ( f (x) ) = g (2x) = 2x + 2

32. Find (f ○ g) (x) = f (g (x) ) = f (x – 5) = 2 (x – 5)² Find (g ○ f ) (x) = g ( f (x) ) = g (2x²) = 2x² - 5

33. 1. f(x) = x² ; g(x) = x + 4 2. f(x) = 4x ; g(x) = x – 3 3. f(x) = 2x ; g(x) = x + 2 4. f(x) = x² ; g(x) = 2x 5. f(x) = x + 1 ; g(x) = 3x = x² + 4 = 4x - 12 = 2x + 4 = 4x² = 3x + 1

34. We 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 = 21

35. Find: 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 = f(1) g(1) = (8(1) + 2) 4(1) = (10) (4) = 40 = f(2) g(2) = 18 8 = 9 4