Final Exam ReviewPages 4-6

Inverses Solving Radical Equations Solving Radical Inequalities Square Root (Domain/Range)

Inverse Concept

• The main concept of an inverse is the x and y coordinates have switched places

( , )x y ( , )y x

FINDING A FORMULA FORAN INVERSE FUNCTION

1. Replace f (x) with y.

2. Interchange x and y.

3. Solve the resulting equation for y.

4. Replace y with f -1(x) if the inverse is a function.

Example: Find the inverse relation algebraically for the function f (x) = 7x + 4.

y = 7x + 4 Original equation defining f

x = 7y + 4 Switch x and y.

7y + 4 = x Reverse sides of the equation.

y = Solve for y.1 4

7 7x

To find the inverse of a relation algebraically, interchange x and y and solve for y.

Find Inverse of f(x)= x2 + 4

• y = x2 + 4 (Swap variables)• x = y2 + 4• y2 = x – 4• y = ±• Don’t forget the (±) when taking a square root

4x

Domain & Range

31 xyWhat’s under the radical determines starting point of the

DOMAIN

What’s after the radical determines starting point of the

RANGE

Domain & Range

31 xy

Domain:

[-1, ∞)

Range:

[3, ∞)

Domain and Range

3 1 5y x

1:[ , )

3D :[5, )R

Graph

2 2 3y x

1 2 63 4 5 7 8 9 10

4

3

2

7

56

8

9

x-axis

y-axis

0

1-2-6 -3-4-5-7-8-910

-4

-3

-2

-1

-7

-5

-6

-8

-9

0

-1

2 2 3y x

Solving Radical Equations

• When solving radical equations make sure only the radical is on the left when you start the problem.

• To UNDO a radical, you take it to the POWER of the Index Number

• ALWAYS Check answers. You may get Extraneous Solutions from radicals.

44 3 1 3 1x x

Solve613 3 x

28x

3 1 3x 1 27x

Solve 4 4 1x

5x Extra

4 3x 4 9x

Solve

362 xx

9x

Solving Radical Inequalities

• MUST put radical expression on left of inequality into graphing calculator as y1 =

• MUST put radical expression on right of inequality into graphing calculator as y2 =

• Graph the equation• Then find INTERSECTION (2nd, TRACE, #5)

Example 1 5 3x

1

2

5

3

y x

y

Now graph and adjust

window to see intersection

Find intersection

2nd Trace #5

Enter Enter Enter

(14, )

Solve

[4,8)

2 4 2 6x

Solve

[2.44, )3 3 5 12x