6.1 radical expression function and models. properties of square roots every positive real number...
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6.1 Radical ExpressionFunction and Models
Properties of Square Roots
Every positive real number has two real-number square roots.
The number 0 has just one square root, 0 itself.
Negative numbers do not have real-number square roots.
Square Root
The number c is a square root of a if c2 = a.
1) Find the square roots of 9Answer: 3 and -3
2) Find the square roots of 36Answer: 6 and -6
3) Find the square roots of 1Answer: 1 and -1
4) Find the square roots of -16 Answer: none because no real number
square gives -16
Examples
principal square root (non-negative
square root)
is the radical sign and a is the radicand
Example 1)
5=
0.2=
-4=
10=
a
100
16
2)5(
04.0
Example
Find principal square roots and simplify.
Slide 5
Examples
More examples6) 7)8)
You may have to use calculator if the radicand is not a perfect square
9) f(x) = Find f(0) = f(3)=
f(-5)=
0
16
11
46 x
=0
= 3.32
406 24
436 22
4)5(6 26430 No real answer
No real answer
10) f(x) = 3x + 12
a) Find f(-1)
= 3(-1) + 12 = -3 + 12 = 9
= 3
b) Find f(x+1)
= 3(x+1) + 12
= 3x + 3 + 12 = 3x + 15
For any real number The principal (nonnegative) square root of a2 is the absolute value of a.
Principle Square Root of a2
2)3()12 b
2)10()11
2)1()14 x
4)13 a
2510)15 2 xx
1010
bb 33 b3
2a 2a
1 x
2)5( x 5 x
Radical Expressions and Functions
Cube Root
Find cube roots, simplifying certain expressions, andfind outputs of cube-root functions.
= 2 because 23 =8
5
6
1.0
x
2
0
y2
For any real number :
a) when k is an even natural number. Use absolute value when k is even unless a is nonnegative.
b) when k is an odd natural number greater than 1. Do not use absolute value when k is odd.
Slide 13
Ex:
1) g(x) = 3 x – 3
Find g(-24)
= 3 -24 – 3 = 3 -27
= -3
2) 5 -32 =
= -2
3) 9 (x + 1)9 =
= x + 1
4) 4 81x4 =
= | 3x |
= 3 | x |
5) Find the domain of
Look at the graph using G.C.
Domain: x + 5 ≥ 0
x≥ -5 [-5, ∞)
5x
6) 2x – 5 - 3
Domain is 2x – 5 ≥ 0
2x ≥ 5
x ≥ 5/2
[5/2, ∞)