1. Function Operations 8.5 8.5 1.Add or subtract functions. 2.Multiply functions. Composite Functions 12.1 1.Find the composition of two functions.

2. Add the following polynomials. 5x + 1 3x 2 – 7x + 6 3x 2 – 2x + 7 f(x) = 5x + 1 (f + g)(x) = = (5x + 1) + (3x 2 – 7x + 6) = 3x 2 – 2x + 7 g(x) = 3x 2 – 7x + 6 f(x) + g(x) Always rewrite!!!

3. Copyright © 2011 Pearson Education, Inc. Adding or Subtracting Functions (f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x).

4. f(x) = 3x + 1 g(x) = 5x + 2 Find: (f + g)(x) (f - g)(x) (g - f)(x) (f - g)(-2) = f(x) + g(x) = (3x + 1) + (5x + 2) = 8x + 3 = f(x) – g(x) = (3x + 1) – (5x + 2) = -2x – 1 = 3x + 1 – 5x – 2 = g(x) – f(x) = (5x + 2) – (3x + 1) = 2x + 1 = 5x + 2 – 3x – 1 = f(-2) – g(-2) f(-2) = 3(-2) + 1 = -5 = (-5) – (-8 ) g(-2)= 5(-2) + 2 = -8 = 3 Always rewrite!!!

5. Slide 3- 5 Copyright © 2011 Pearson Education, Inc. Given f(x) = 4x – 1 and g(x) = 5x + 2, what is (f + g)(x)? a) x + 4 b) x − 4 c) 9x + 1 d) 9x – 1 8.5

6. Slide 3- 6 Copyright © 2011 Pearson Education, Inc. Given f(x) = 4x – 1 and g(x) = 5x + 2, what is (f + g)(x)? a) x + 4 b) x − 4 c) 9x + 1 d) 9x – 1 8.5

7. Multiplying Functions (f g)(x) = f(x) ∙ g(x).

8. Copyright © 2011 Pearson Education, Inc. f(x) = 2x + 7 and g(x) = x − 4 Find (f g)(x). = (2x + 7)(x − 4) = 2x 2 − 8x + 7x – 28 = 2x 2 − x – 28 (f g)(x) = f(x)∙g(x) Always rewrite!!!

9. f(x) = – x 2 – 8x + 2 g(x) = x + 2 h(x) = x – 8 Find: (gh)(x) (fg)(0) (fh)(-1) (f h)(x) =g(x) ∙ h(x) = (x + 2)(x – 8) = x 2 – 6x - 16 = f(0) ∙ g(0) = (2)(2) = 4 = f(-1) ∙ h(-1) = (9)(-9) = -81 = f(x) ∙ h(x) = (-x 2 – 8x + 2)(x – 8) = -x 3 + 66x – 16 f(-1) = -(-1) 2 – 8(-1) + 2 = -1 + 8 + 2 = 9

10. Slide 3- 10 Copyright © 2011 Pearson Education, Inc. Given f(x) = 3x – 2 and g(x) = 5x – 1, what is (f g)(x)? a) 15x 2 − 13x + 2 b) 15x 2 − 13x − 2 c) 15x 2 − 7x + 2 d) 15x 2 − 7x − 2 8.5

11. Slide 3- 11 Copyright © 2011 Pearson Education, Inc. Given f(x) = 3x – 2 and g(x) = 5x – 1, what is (f g)(x)? a) 15x 2 − 13x + 2 b) 15x 2 − 13x − 2 c) 15x 2 − 7x + 2 d) 15x 2 − 7x − 2 8.5

12. f(x) = 2x + 3 g(x) = x + 4 f (2) = f (a) = f (x+4) = f (g(x)) = 2(2) + 3 = 7 2a + 3 2(x + 4) + 3 = Composition of Functions (f ◦ g)(x) = Nested Format 2x + 8 + 3 =2x + 11

13. Composition of Functions Shorthand notation for substitution. Nested Format Always rewrite composition of functions in nested format! Read “f of g of x”.

14. If and find. Find f(1). Simplify. Substitute 1 for g(3) Find g(3). Write in nested format. g(3) = 2(3) – 5 = 1

15. f(x) = x 2 – 8x + 2 g(x) = x + 2 h(x) = x – 8 Find: = g(h(3)) h(3) = 3 – 8 = -5 = g(-5) = -3 = h(f(x)) = h(x 2 – 8x + 2) = (x 2 – 8x + 2) - 8 = f(g(x)) = f(x + 2) = (x + 2) 2 – 8(x + 2) + 2 = x 2 – 8x – 6 = x 2 + 4x + 4 – 8x – 16 + 2 = x 2 – 4x – 10 (x + 2) 2 (x + 2)(x + 2) x 2 + 4x + 4 (x + 2) 2 x 2 + 4 X g(-5) = -5 + 2 = -3 Rewrite & Foil Always rewrite!!!

16. Slide 12- 16 Copyright © 2011 Pearson Education, Inc. If f(x) = x + 7 and g(x) = 2x – 12, what is a)  44 b)  3 c) 3 d) 44

17. Slide 12- 17 Copyright © 2011 Pearson Education, Inc. If f(x) = x + 7 and g(x) = 2x – 12, what is a)  44 b)  3 c) 3 d) 44