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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