1. Rational Exponents and Radical Functions
Unit 6
Rational Exponents and Radical Functions

2. Unit Essential Question:
What are the different properties of exponents and radicals that allow us to solve equations?

3. Rational Exponents and Nth Roots
Lesson 6.1
Rational Exponents and Nth Roots

4. Lesson Essential Question:
How are nth roots and rational exponents related and how can they be used to simplify expressions?

5. Nth Roots
How do we know how many answers there will be for simplifying nth roots?
𝑛 𝑎 = ?

6. Nth Roots 𝑛 𝑎 If n is even: When a < 0, there are no real roots.
When a > 0, there are two real roots.
If n is odd:
When a < 0, there is one real root.
When a > 0, there is one real root.

7. Rational Exponents: How can we rewrite nth roots as exponents?
How can we rewrite exponents as nth roots?

8. Homework:
Pages #’s 3 – 49 odds

9. Bell Work: Simplify by hand (no calculators): 1) 8 2 3 2) (−27) 4 31)
2) (−27) 4 3
3) 36 − 3 2

10. Examples: Convert each expression into radical form (DO NOT SIMPLIFY):
5 3 2
(−18) 3 5
9 − 3 4

11. Examples:
Convert each radical into exponential form:
3 7
6 3 7
5 𝑥 3

12. Apply Properties of Rational Exponents
Lesson 6.2
Apply Properties of Rational Exponents

13. Lesson Essential Question:
What are the different properties of rational exponents and how are they used to simplify expressions?

14. Properties of Rational Exponents:
Blue Tables on Pages 420 – 421

15. Examples:

16. Bell Work: Simplify without using a calculator: 1) (49) 3/2
1) (49) 3/2
2) 𝑥 8 𝑦 11
3) 8 − 3 4

17. Homework:
Pages 424 – 425 #’s 3 – 65 odds

18. Bell Work: Simplify each expression completely: 1) 8 3 4 ∙ 8 5 4
1) ∙
2) ∙
3) 𝑥 9 𝑦 12 𝑧 10 ∙ 5 9𝑥𝑦

19. Class Work: Pages 424 – 425 #’s 4 – 54 evens
This assignment will be collected!!!
Small Quiz tomorrow at the beginning of class on Properties of Exponents and Radicals. (NO CALCULATORS!!!)

20. Bell Work:
1) (−243) 3/5 2) ) 𝑥 3 𝑦 9 𝑧 14

21. Quiz:

22. Operations and compositions of functions
Lesson 6.3
Operations and compositions of functions

23. Lesson Essential Question:
How do we perform operations on functions, and how do we take a composition of two or more functions?

24. Operations on Functions:
If 𝑓 𝑥 = 𝑥 2 −2𝑥 𝑎𝑛𝑑 𝑔 𝑥 =3𝑥−6, then we can perform the following operations:
1) 𝑓 𝑥 +𝑔 𝑥
2) 𝑓 𝑥 −𝑔 𝑥
3) 𝑓 𝑥 ∙𝑔 𝑥
4) 𝑓 𝑥 𝑔 𝑥

25. Domain:
Remember that domain is the set of all x-values that will work for a given function. When we perform operations, this can sometimes alter the domain of the original functions.
Lets look back at the four operations we did, and find the domain for each:

26. Compositions of Functions:
A composition is when we use one function to evaluate another. A new function or “composition” is composed of two more other functions that are joined together.
Example: 𝐼𝑓 𝑓 𝑥 =3𝑥−1 𝑎𝑛𝑑 𝑔 𝑥 = 𝑥 2 +5:
𝑇ℎ𝑒𝑛 𝑓𝑖𝑛𝑑 𝑔 𝑓 𝑥 𝑎𝑛𝑑 𝑓 𝑔 𝑥 .

27. Notation: There are two different ways to write compositions.
𝑓 𝑔 𝑥 =𝑓 𝑜 𝑔, the “o” means that f is a function of g.

28. Homework:
Page 432 #’s 3 – 37 odds

29. Bell Work: 𝐼𝑓 𝑓 𝑥 =4 𝑥 1/2 𝑎𝑛𝑑 𝑔 𝑥 =16 𝑥 1/2 , 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔:
𝐼𝑓 𝑓 𝑥 =4 𝑥 1/2 𝑎𝑛𝑑 𝑔 𝑥 =16 𝑥 1/2 , 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔:
1) f(x) + g(x)
2) f(x) – g(x)
3) f(x) ▪ g(x)
4) f(x) / g(x)
5) f(g(x))
6) g(f(x))
7) Find the domain of 1 – 6.

30. Classwork/Homework: This assignment will be collected tomorrow!!!
Pages 432 – 433 #’s 4 – 38 evens

31. Bell Work: 1) If 𝑓 𝑥 =3𝑥−2 and 𝑔 𝑥 = 𝑥+2 3 , find 𝑓 𝑔 𝑥 .

32. Lesson 6.4
Inverse functions

33. Lesson Essential Question:
How do we find an inverse function and how does it relate to its original function???

34. Inverse Functions:
To prove that two functions are inverses, then the composition of the two functions must come out to be x.
If f(x) and g(x) are inverses, then f(g(x)) = x and g(f(x)) = x.

35. Graphs of Inverse Functions:
If we look at the graphs of inverse functions, what do you see?
Ex: Graph 𝑓 𝑥 =3𝑥+1 and 𝑔 𝑥 = 𝑥−1 3
Then lets look at:
Ex: Graph 𝑓 𝑥 = 2𝑥 3 and 𝑔 𝑥 = 𝑥

36. If two functions are inverses, then…
1) The composition of the functions will come out to be x.
2) They have reversed x and y values. (This means the domain and range are switched!)
3) The graphs of the functions are reflected about the line y = x.

37. Notation:
If we are solving to find an inverse based upon the original function 𝑓(𝑥), then its inverse is called 𝑓 −1 𝑥 .
Lets find some inverses!!! YEEEEHAAAWWWW!

38. Bell Work: Find the inverses for the following functions: 1) 𝑓 𝑥 =5𝑥−4
1) 𝑓 𝑥 =5𝑥−4
2) 𝑓 𝑥 = 3𝑥 3 −8
3) 𝑓 𝑥 = 𝑥+3
4) 𝑓 𝑥 = 2𝑥−1 𝑥+3

39. Steps for finding an inverse function…
1) All x’s and y’s switch.
2) Solve the new equation for y.
3) Rewrite the new function as 𝑓 −1 𝑥 .
4) Use either a composition or graphs to prove you found the right inverse.

40. Homework:
Pages 442 – 443 #’s 3 – 27 odds

41. Bell Work:
1) Using a composition, prove that 𝑓 𝑥 = 2𝑥 2 −9 and 𝑔 𝑥 = 𝑥 are inverses.
2) Find the inverse of ℎ 𝑥 = 4𝑥+3 12
3) Find the inverse of 𝑓 𝑥 = 3 𝑥−3
4) Find the inverse of 𝑔 𝑥 = 𝑥−6 𝑥+2

42. Solving Radical Equations
Lesson 6.6
Solving Radical Equations

43. Lesson Essential Question:
How do we solve radical equations and why is it important to check our solutions?

44. Steps for solving a radical equation:
1) Isolate one of the radicals
2) Raise both sides to a power to cancel out the isolated radical.
3) Repeat steps 1 and 2 if necessary.
4) Solve.
5) Check for extraneous solutions!

45. Examples: Solve each equation: 1) 4𝑥−8 =10 2) 5−2𝑥 =7 3) 3 3𝑥−7 =2
1) 4𝑥−8 =10
2) 5−2𝑥 =7
3) 3 3𝑥−7 =2
4) 𝑥+5= −2𝑥−2
5) 24−10𝑥 =𝑥−4

46. Examples:
𝑥+2 +1= 3−𝑥
5𝑥+10=4𝑥+9+ 7𝑥+15
𝑥+6 =2+ 𝑥−2

47. Homework:
Pages 456 – 457 #’s 3 – 21 odds and 35 – 51 odds

48. Bell Work: Solve: 1) 10𝑥+9 =𝑥+3 2) 9𝑥 3/5 =72 3) 2𝑥+9 = 𝑥+7
1) 10𝑥+9 =𝑥+3
2) 9𝑥 3/5 =72
3) 2𝑥+9 = 𝑥+7
4) 2𝑥−3 − 𝑥+7 +2=0

49. More Examples!!! Sweeeeeet…
5) 3 2𝑥− −𝑥 =11
6) 7−2𝑥 − 5+𝑥 = 4+3𝑥
7) 𝑥+1 = 3𝑥−5
8) 𝑥 = 𝑥 +1

50. Homework: This assignment will be collected!!!
Pages 456 – 457 #’s 4, 10, 16, 20, 24, 28, 38, 50, 52

51. Bell Work: 1) What is the inverse function of 𝑓 𝑥 = 𝑥+4 2𝑥−1 .
2) Solve: 𝑥=3+ 5𝑥−9

52. Unit 6 Test Upcoming!!!!!!!!!!!!!!!!!!!!!!! Properties of Radicals
Properties of Exponents
Operations and Compositions of Functions
Inverse Functions
Solving Radical Equations