6.1 D o N o/w

1. 6.1 DoNo/w We know that We also know that (i.e. 52*53=55) If 5x*5x=51, what do you think x would have to be? If 5x*5x*5x=51 what do you think x would have to be? What conclusions can you make?

2. nth Roots! for any integer n greater than 1.

3. Finding Real nth Roots • Evaluate:

4. Find the Indicated Real nth Root(s) of a

5. Does the numerator have to be 1? • Find • Well we know this is (2)2=4.

6. Key Concept!

8. Evaluate expressions without using a calculator. 1. 813/4 2. 17/8

9. Evaluate expressions without using a calculator. 3. 45/2 4. 9–1/2

10. Solving Equations! • Solve. • 4x5=128 • (x+3)4=362

11. PRACTICE PROBLEMS!

12. 6.2 Do Now • Simplify 52*58. • Evaluate .

13. Do the same rules for integer exponents apply? • Evaluate • Now realize that this is

15. Simplify 4.

16. Radical Notation for Properties 3 and 6! Simplify: a) b)

17. If only life were so simple… • Simplest radical form: rationalize the denominator and take out any perfect nth powers. • If the index and radicand are the same, we have like radicals, and we can add or subtract these.

18. Examples: Write in Simplest Radical Form 1) 2) 3)

19. PRACTICE! • Write in simplest radical form.

20. With Variables! • Why do we generally assume all variables are positive? Taking roots when n is odd and the radicand is negative, gives an odd result.

21. Examples • Write in simplest form—assume all variables are positive.

22. Practice

23. Application Presentation Problem

24. 6.3 Do ο Now • You are shopping for spectacles. You find a pair for $36. You have a $5 off coupon, but you also notice that the store is offering a 25% off discount. • You don’t want to make a spectacle of yourself, but you wonder which discount will be applied first. Find the price of the spectacles if the $5 coupon is used first followed by the 25% off discount. Then find the price if the 25% discount is applied first, followed by the $5 discount. Is the price the same?

25. Conjunction Function, Mix Those Functions • Remind me what a function is….in your own words!

26. Power function: y=axb where a is a real number and b is a rational number.

27. Example: Add and Subtract • Suppose f(x)=5x1/3 and g(x)=-11x1/3. • (a) Find f(x)+g(x). • (b) Find f(x)-g(x). • (c)Find the domains of f+g and f-g

28. Example: Multiply and Divide • Suppose f(x)=x2+1 and g(x)=x1/2. • (a) Find f(x)*g(x). • (b) Find f(x)/g(x). • (c)Find the domains of f*g and f/g.

29. YOU TRY! • Suppose • Find f*g and g/h, as well as the domain of each.

30. Back to the Do Now

31. What operation are we applying to f(x) and g(x) to get those new functions? • Hmmm….need something new!

32. How does it work? Start inside! • f(g(2)) • g(f(2)) • f(f(2)) • g(h(4))= • h(f(-3))

33. With Functions

34. Back to the Spectacles! • How can we represent the discounts using composition of functions?

35. You Try!

36. 6.4 (Do Now)-1 • Solve y=2x+3 for x.

37. f( )= f-1( )=

38. Inverse Functions • Consider f(x)=2x+3. • The inverse function goes the other way. The inverse of f(x)=2x+3 is f=1(y)=(y-3)/2. Notice that the inverse is a function of y! The range of the function is the domain of its inverse and vice-versa.

39. Again, if f(x)=2x+3, We can write this as:

40. Find the inverse of f(x)=5x+2 How can we check our result?

41. Try This! • Find the inverse of

42. How to Verify

43. You Try! • Verify that f(x)=x-5 and g(x)=-x-5 are inverses.

45. YOU TRY! Find the inverse of the function, and use it to find the number of dollars that could be obtained for 250 euros on that day.

46. (Do Now)-1 Which of the following are functions? Justify your answer. A B C D

47. Inverse • Let’s use our procedure to see what the inverse of f(x)=x2 should be:

48. Graph

49. Restricting the Domain • For which values of x does f(x)=x2 actually have an inverse? • If we take only nonnegative real numbers, then it will have an inverse. Let’s find the inverse for

50. Similarly • What about g(x)=x3? • In this case, is a function, and so we do not have to restrict the domain.