1. Chapter 7
Radical Equations

2. Operations of Functions
Lesson 7.1
Operations of Functions

3. Operations with functions
pg 1
Operations with functions
(f + g)(x)= f(x) + g(x)
(f – g )(x) = f(x) – g(x)
(f · g)(x) = f(x) · g(x)
(f/g)(x) = f(x) , g(x) = 0
g(x)

4. Examples 1. f(x)= x2 – 3x + 1 and g(x) = 4x + 5 find
pg 1
Examples
1. f(x)= x2 – 3x + 1 and g(x) = 4x + 5 find
a. (f + g)(x) b. (f – g)(x)

5. 2. f(x) = x2 + 5x – 1 and g(x) = 3x – 2 find
pg 1
2. f(x) = x2 + 5x – 1 and g(x) = 3x – 2 find
a. (f · g)(x) b. (f/g)(x)

6. Composition of Functions
pg 1
Composition of Functions
[f o g](x) = f[g(x)]
Plug in the full function of g for x in function f

7. Examples 3. Find [f o g](x) and [g o f](x)
pg 1
Examples
3. Find [f o g](x) and [g o f](x)
for f(x) = x+3 and g(x) = x2 + x – 1
4. Evaluate [f o g](x) and [g o f](x) for x = 2

8. TOD: Find the sum, difference, product and quotient of f(x) and g(x)
1. f(x) = x2 + 3
g(x) = x – 4
Find [g o h](x) and [h o g](x)
2. g(x) = 2x h(x) = 3x – 4
If f(x) = 3x, g(x) = x + 7 and h(x) = x2 find the following
4. f[g(3)] 5. g[h(-2)] 6. h[h(1)]

9. Inverse Functions and Relations
Lesson 7.2
Inverse Functions and Relations

10. pg 2
Inverse Relations
Two relations are inverse relations if and only if one relation contains the element (a, b) the other relation contains the element (b, a)
Example: Q and S are inverse relations
Q = {(1, 2), (3, 4), (5, 6)}
S = {(2, 1), (4, 3), (6, 5)}

11. Examples 1. Find the inverse relation of {(2,1), (5,1), (2,-4)}
pg 2
Examples
1. Find the inverse relation of {(2,1), (5,1), (2,-4)}
2. Find the inverse relation of {(-8,-3), (-8,-6), (-3,-6)}

12. Property of Inverse Functions
pg 2
Property of Inverse Functions
Suppose f and f-1 are inverse functions. Then f(a) = b if and only if f -1 (b) = a
To find the inverse of a function:
1. Replace f(x) with y
2. Switch x and y
3. Solve for y
4. Replace y with f-1(x)
5. Graph f(x) and f-1(x) on the same coordinate plane

13. To Graph a function and it’s inverse
pg 2
To Graph a function and it’s inverse
1. Make an x/y chart for f(x) then graph the points and connect the dots
2. Make an x/y chart for f-1(x) by switching the x and y coordinates and then graph and connect the dots
- The graphs should be reflections of one another over the line y=x

14. Examples
3. Find the inverse of and then graph the function and its inverse
pg 3

15. pg 3
4. Find the inverse of and then graph the function and its inverse f(x) = 2x - 3

16. pg 3
Inverse Functions
Two functions f and g are inverse functions if and only if both of their compositions are the identity function.
[f o g](x) = x and [g o f](x) = x

17. 5. Determine if the functions are inverses
pg 3
5. Determine if the functions are inverses

18. 6. Determine if the functions are inverses
pg 3
6. Determine if the functions are inverses

19. Square Root Functions and Inequalities
Lesson 7.3
Square Root Functions and Inequalities

20. Square Root Functions
Definition: when a function contains a square root of a variable
The inverse of a quadratic function (starts with x2) is a square root function only if the range has no negative numbers!!
pg 4

21. Parent Functions
pg 4

22. Graphing Square Root Functions
pg 4
Graphing Square Root Functions
1. Find the domain.
The radicand (the stuff inside the square root) cannot be negative so take whatever is inside and make it greater than or equal to 0 and solve.
2. Plug the x value you found back in and solve for y.
3. Make a table starting with the ordered pair you found in steps 1 & 2. Graph.
4. State the range.

23. Examples 1. Graph. State the domain, range, and x- and y- intercepts.
pg 5

24. 2. Graph. State the domain, range, and x- and y- intercepts.
pg 5

25. 3. Graph. State the domain, range, and x- and y- intercepts
pg 5

26. Square Root Inequalities
pg 6
Square Root Inequalities
Follow same steps as an equation to graph but add last step of shading.
Remember rules for solid and dotted lines!!

27. Examples
4. Graph
pg 6

28. 5. Graph
pg 6

29. 6. Graph
pg 6

30. 7. Graph
pg 6

31. 7.4
Nth Roots

32. pg 7
Nth Roots

33. Symbols and Vocabulary
pg 7
Symbols and Vocabulary
Radical Sign
index
Radicand

34. More Vocabulary Principal Root: the nonnegative root
Example: 36 has two square roots, 6 and -6
6 is the principal root because it is positive
Other things to remember:
- If the radical sign has a – in front of it this indicates the opposite of the principal square root
- If the radical has a ± in front of it then you give both the principal and the opposite principal roots
pg 7

35. Summary of Nth Roots n b > 0 b < 0 b = 0 even
one positive root, one negative root
no real roots
One real root = 0
odd
one positive root, no negative root
no positive roots, one negative root
pg 7

36. Examples
pg 8

37. Your turn…
pg 8

38. More things to remember…
When you find the nth root of an even power and the result is an odd power, you must take the absolute value of the result to ensure that the answer is nonnegative
(− 5) 2 = −5 𝑜𝑟 5
(− 2) 6 = (−2) 3 𝑜𝑟 8
pg 8

39. Examples Using Absolute Values
pg 8

40. Operations with Radical Expressions
7.5
Operations with Radical Expressions

41. Review of Properties of Radicals
Product
Property
If all parts of the radicand are positive- separate each part so that it has the nth root
Ex:
Quotient Property
pg 9

42. When is a radical simplified?
pg 9
When is a radical simplified?
The nth root is as small as possible
The radicand has no fractions
There are no radicals in the denominator

43. Rationalizing the Denominator
If you have a fraction with a radical in the denominator you must rationalize the denominator, multiply the numerator and denominator by the square root in the denominator.
pg 9

44. Examples
pg 9

45. pg 10
Simplify all radicals then combine -Remember you can only combine like terms!

46. pg 10
Foil- then simplify

47. Multiply by the conjugate- then simplify
pg 10
Multiply by the conjugate- then simplify

48. 7.6
Rational Exponents

49. Fractions, Fractions, are our Friends!
What do we do when we have a fraction as an exponent?
Change it into a radical
The denominator of becomes the index of the radical
The numerator becomes the power for the radicand
Ex: =
pg 11

50. Examples
pg 11

51. Generalization for numerator greater than 1
The denominator becomes the index for the radical (nth root)
If the numerator is bigger than 1:
The numerator becomes the power for the nth root of the radicand
Ex: = = =
pg 11

52. pg 12
Examples

53. pg 12

54. pg 12

55. Solving Radical Equations and Inequalities
7.7
Solving Radical Equations and Inequalities

56. pg 13
Vocabulary
Radical equations/inequalities: equations/inequalities that have variables in the radicands
Extraneous Solution: when you get a solution that does not satisfy the original equation.

57. Steps to solve radical equations/inequalities
pg 13
Steps to solve radical equations/inequalities
1. Isolate the radical.
If there is more than one, isolate the radical that has the most stuff in it first!
2. Raise both sides to the power that will eliminate the radical.
If there is more than one radical you will have to repeat this step until there are no more radicals in the problem
3. Solve for the variable.
4. Test solutions to check for extraneous roots.

58. Solve Radical Equations
pg 13
Solve Radical Equations

59. pg 13

60. pg 14

61. pg 14

62. Day #2 Now let’s look at inequalities
pg 14
Day #2 Now let’s look at inequalities
Steps to Solve a radical inequality
1. Identify the values of x for which the root is defined.
2. Solve given inequality by isolating and then eliminating the radical
3.Test values to confirm your solution (use a table to do this)
4. Graph solution on a number line.
(Remember open and closed dots)

63. pg 14

64. pg 14

65. pg 14