1. Section 1.8 Combinations of Functions: Composite Functions

2. 1st Day Sum, Difference, Product, and Quotient of Functions

3. Let f and g be two functions with overlapping domains. Then, for all x common to both domains, the sum, difference, product, and quotient of f and g are defined as follows. • 1. Sum: (f + g)(x) = f (x) + g(x) • 2. Difference: (f − g)(x) = f (x) − g(x) • 3. Product: (fg)(x) = f (x) ∙ g(x) • 4. Quotient:

4. Example 1 • Find (a) (f + g)(x), (b) (f – g)(x), (c) (fg)(x), and • (d) What is the domain of

5. 1) f (x) = 2x + 5 and g(x) = 2 – x • a) (f + g)(x) = f (x) + g(x) • = (2x + 5) + (2 – x) • = (2x – x) + (5 + 2) • = x + 7 • b) (f − g)(x) = f (x) − g(x) • = (2x + 5) − (2 – x) • = (2x+ x) + (5 − 2) • = 3x+ 3

6. 1) f (x) = 2x + 5 and g(x) = 2 – x • c) (f g)(x) = f (x) ∙ g(x) • = (2x + 5)(2 – x) • = 4x – 2x2 + 10 − 5x • = −2x2 − x + 10

7. 2) f (x) = 3x – 2 and g(x) = x + 7 • a) (f + g)(x) = f (x) + g(x) • = (3x − 2) + (x + 7) • = (3x + x) + (−2 + 7) • = 4x+ 5 • b) (f − g)(x) = f (x) − g(x) • = (3x − 2) − (x + 7) • = (3x − x) + (−2 − 7) • = 2x − 9

8. 2) f (x) = 3x − 2 and g(x) = x + 7 • c) (f g)(x) = f (x) ∙ g(x) • = (3x − 2)(x + 7) • = 3x2 + 21x − 2x − 14 • = 3x2 + 19x − 14

9. a) (f + g)(x) = f (x) + g(x) b) (f − g)(x) = f (x) − g(x)

10. c) (fg)(x) = f (x) ∙ g(x)

12. Example 2 • Evaluate the indicated function for f (x) = x2 +1 and g(x) = x – 4. • 1) (f – g)(−1) • (f – g)(−1) = f (−1) − g(−1) • = [(−1)2 + 1] − [−1 − 4] • = 2 − (−5) • = 7

13. 2) (fg)(5) + f (4) • (fg)(5) + f (4)= f (5) ∙ g(5) + f (4) • = [(5)2 + 1][5 − 4] + [(4)2 + 1] • = (26)(1) + 17 • = 26 + 17 • = 43

14. 3) (f + g)(t – 2) • (f + g)(t – 2) = f (t − 2) + g(t− 2) • = [(t − 2)2 + 1] + [(t − 2) − 4] • = [t2 − 4t + 4 + 1] + [t − 6] • = t2 – 4t + 5 + t – 6 • = t2 – 3t – 1 HW: p. 89 (5-23 odd)

15. 2nd Day • The composition of the function fwith the function g is: • (f ◦ g)(x) = f (g(x)) • The domain of the composition is the set of all x in the domain of g.

16. Example 1 • Given f (x) = x + 2 and g(x) = 4 – x2. Find: • a) (f ◦ g)(x) • b) (g ◦ f )(x) • c) (g ◦ f )(−2)

17. Given f (x) = x + 2 and g(x) = 4 – x2. • a) (f ◦ g)(x) • (f ◦ g)(x) = f (g(x)) • = f (4 – x2) • = [4 – x2] + 2 • = 6 – x2

18. 1) Given f (x) = x + 2 and g(x) = 4 – x2. • b) (g ◦ f )(x) • (g ◦ f ) (x) = g (f (x)) • = g(x + 2) • = 4 – (x + 2)2 • = 4 – [x2 + 4x + 4] • = 4 – x2 – 4x – 4 • = –x2– 4x • c) (g ◦ f )(−2) = –(−2)2– 4(−2) • = 4

19. Example 2 • Given f (x) = 3x + 5 and g(x) = 5 – x. Find: • a) (f ◦ g)(x) • b) (g ◦ f )(x) • c) (f ◦ f )(x)

20. Given f (x) = 3x + 5 and g(x) = 5 – x. • a) (f ◦ g)(x) • (f ◦ g)(x) = f (g(x)) • = f (5 – x) • = 3[5 – x] + 5 • = 15 – 3x + 5 • = 20 – 3x

21. Given f (x) = 3x + 5 and g(x) = 5 – x. • b) (g ◦ f )(x) • (g ◦ f ) (x) = g (f (x)) • = g(3x + 5) • = 5 – (3x + 5) • = 4 – 3x – 5 • =–3x – 1

22. Given f (x) = 3x + 5 and g(x) = 5 – x. • c) (f ◦ f )(x) • (f ◦ f ) (x) = f (f (x)) • = f (3x + 5) • = 3(3x + 5) + 5 • = 9x + 15 + 5 • = 9x + 20

23. Example 3 • Given: f (x) = x2 – 9 and • find the composition (f ◦ g)(x). Then find the domain of (f ◦ g)(x).

24. Given: f (x) = x2 – 9 and • (f ◦ g)(x) = f (g(x)) • The domain of (f ◦ g)(x) is the domain of g(x), • which is [−3, 3].

26. Example 4 • Given: and g(x) = x + 1, find • (a) (f ◦ g)(x) • (b) (g ◦ f )(x) • Find the domain of each function and each composite function.

27. Domain of f (x): • Domain of g(x):

28. Given: and g(x) = x + 1, find • (a) (f ◦ g)(x) • (f ◦ g)(x) = f (g(x)) • = f (x + 1)

29. Domain of (f ◦ g)(x) :

30. Given: and g(x) = x + 1, find • (b) (g ◦ f )(x) • (g ◦ f )(x) = g( f (x)) • Domain: HW: pp. 89-90 (31-41 odd)