Warm-up for Section 3.2:

1. Warm-up for Section 3.2:

2. Warm-up for Section 3.2:

3. 3.1B Homework Answers 28 = 256 2. (-7)3 = -343 3. 1/47 = 1/16384 4. 1/54= 1/625 5. 1/44= 1/256 6. 1/86 = 1/262,144 7. 1.342 10128. 3.38 10-59. 2.025  109 3.73248 10-7 11. 6.6 10112. 3.5  10-15 13. x414. y1115. 531,441x18 16. 17. 18.

4. 3.1B Homework Answers Continued… 19. 20. 21. 22. 4q3r 23. 24. 25. 26. 27. 1.032 1011

5. Operations on Functions Section 3.2B Standard: MM2A5d Essential Question: How do I perform operations with functions?

6. Vocabulary • Power function: a function of the form y = axb,wherea is a real number and b is a rational number • Composition: h(x) = g(f(x)) is the composition of a function g with a function f. • The domain of h is the set of all x-values such that x is in the domain of f and f(x) is the domain of g.

7. Investigation 1: New functions can be created from established functions through the operations of addition, subtraction, multiplication, and division. Consider the linear function f(x) = 2x + 1 and the quadratic function g(x) = x2 – 3. Complete the table below for the selected values of the domain. The first column has been done for you. Table 1:

8. Table 1: -1 1 7 -3 6 -2 Now, keeping the domain fixed, add the range values for f and g to create a new function. Complete the table below to identify the y values for this new function. The first column has been done for you. Table 2: -3 -2 13

9. This new function is denoted y = f(x) + g(x) or y = (f + g)(x). To find the rule for the new function, simply add the expressions for y = f(x) and y = g(x). This new function is: y = (2x + 1) + (x2 – 3). In simple form, we have y = x2 + 2x – 2. Let’s call this function h. So, h(x) = x2 + 2x – 2. Evaluate the function for each domain element to check the values in Table 2.

10. Let’s call this function h. So, h(x) = x2 + 2x – 2. Evaluate the function for each domain element to check the values in Table 2. h(-2) = (-2)2 + 2(-2) – 2 = _______ h(-1) = (-1)2 + 2(-1) – 2 = _______ h(0) = (0)2 + 2(0) – 2 = _______ h(3) = (3)2 + 2(3) – 2 = _______ Did you get the same values in Table 2? ________ -2 -3 -2 13 YES

11. Let’s use f(x) = 2x + 1 and g(x) = x2 – 3 and the operations of subtraction, multiplication, and division to create other new functions. Call the new functions s(x), m(x), and d(x), respectively. • (2). f(x) – g(x) or (f – g)(x) • = (2x + 1) – (x2 – 3) • = 2x + 1 – x2 + 3 • = -x2 + 2x + 4 • s(x) = -x2 + 2x + 4

12. Let’s use f(x) = 2x + 1 and g(x) = x2 – 3 and the operations of subtraction, multiplication, and division to create other new functions. Call the new functions s(x), m(x), and d(x), respectively. • (3). f(x) ∙ g(x) or (fg)(x) • = (2x + 1)(x2 – 3) • = 2x3 – 6x + x2 – 3 • = 2x3 + x2 – 6x – 3 • m(x) = 2x3 + x2 – 6x – 3

13. Let’s use f(x) = 2x + 1 and g(x) = x2 – 3 and the operations of subtraction, multiplication, and division to create other new functions. Call the new functions s(x), m(x), and d(x), respectively. • (4). or

14. The domain of the new function is the set of values common to original functions. In other words, it is the intersection of the domains of the original functions. The domain of f(x) = 2x + 1 is _____________ and the domain of g(x) = x2 – 3 is ___________. So, the domains for h, s, and m will all be ___________. all reals all reals all reals

15.  But, the function d was created by division so we must check to see what values of the common domain will make the denominator zero. This value must be excluded. So, the domain of y = d(x) is all reals except __________. x = -½ 2x + 1 = 0 2x = -1 x = -1/2

16. Check for Understanding: Let h(x) = 3x + 1 and p(x) = 2x – 5 Find the following and state the domain. (5). h(x) + p(x) or (h + p)(x) = ____________ Domain: _____________ (6). h(x) – p(x) or (h – p)(x) = ___________ Domain: _____________ 5x – 4 all reals x + 6 all reals

17. Check for Understanding: Let h(x) = 3x + 1 and p(x) = 2x – 5 Find the following and state the domain. (7). h(x) ∙ p(x) or (hp)(x) = ____________ Domain: _____________ (8). or = ___________ Domain: ______________________ 6x2 – 13x – 5 all reals all reals except x = 5/2

18. Another way of combining two functions is to form the composition of one with the other. g(x) = x2 f(x) = x + 1 Df Rf -2 3 4 -1 4 5 1 16 25 Dg Rg The composition of g with f can be pictured above.

19. The new function created maps the domain of f to the range of g. g(x) = x2 f(x) = x + 1 Df Rf -2 3 4 -1 4 5 1 16 25 Rg Dg If we call this new function h, then the rule for h is h(x) = (x + 1)2

20. The domain of h is the set of all x-values such That x is in the domain of g and g(x) is in the domain of f. h(x) = f(g(x))

21. (9). Let f(x) = 6x and g(x) = 3x + 5 find each composition and its domain. a. f(g(x)) = Domain: _________ b. g(f(x)) = Domain: _________ f(3x + 5) = 6(3x + 5) all reals h(x) = 18x + 5 g(6x) = 3(6x) + 5 all reals h(x) = 18x + 5

22. (9). Let f(x) = 6x and g(x) = 3x + 5 find each composition and its domain. c. f(f(x)) = Domain: __________ d. g(g(x)) = Domain: _________ f(6x) = 6(6x) all reals h(x) = 36x g(3x + 5) = 3(3x + 5) + 5 = 9x + 15 + 5 all reals h(x) = 9x + 20

23. (10). Let f(x) = 2x and g(x) = x2 – 3 find each composition and its domain. a. f(g(x)) = Domain = __________ b. g(f(x) = Domian = _________ f(x2 – 3 ) = 2(x2 – 3 ) h(x) = 2x2 – 6 all reals g(2x) = (2x)2 – 3 h(x) = 4x2 – 3 all reals

24. (10). Let f(x) = 2x and g(x) = x2 – 3 find each composition and its domain. c. g(g(x)) = Domian = _________ g(x2 – 3 ) = (x2 – 3)2 – 3 = (x2 – 3)(x2 – 3) – 3 = x4 – 3x2 – 3x2 + 9 – 3 h(x) = x4 – 6x2 + 6 all reals

25. Check for Understanding: Let p(x) = 3x + 1 and h(x) = x2 – 4, find each new functions and its domain. (11). (p + h)(x) = _____________________ Domain: ________________ (12). (h – p)(x) = _____________________ Domain: ________________ (13). (ph)(x) = _____________________ Domain: ________________ x2 + 3x – 3 all reals x2 – 3x – 5 all reals 3x3 + x2 – 12x – 4 all reals

26. (14). = _____________________ Domain: ___________________  (15). p(h(x)) = _____________________ Domain: ________________ (16). h(p(x)) = _____________________ Domain: ________________ all reals except x = ±2 3x2 – 11 all reals 9x2 + 6x – 3 all reals