Radical Functions
Download
Report
Transcript Radical Functions
Chapter 7
Radicals,
Radical Functions, and
Rational Exponents
7.1 Radical Expressions and
Functions
Square Root
If a >= 0,
then b >= 0, such that b2 = a, is the principal square
root of a
√a=b
E.g.,
√25 = 5
√100 = 10
4
2
2 2 4
---- = ----, because --- = ----49
7
7
49
9 + 16
= 25 = 5
9 + 16 = 3 + 4 = 7
Negative Square Root
25 = 5 ---- principal square root
- 25 = -5 ---- negative square root
Given: a
What is the square root of a?
Given: 25
What is the square root of 25?
sqrt = 5, sqrt = -5, because 52 = 25, (-5)2 = 25
Square Root Function
f(x) = x
y
x
y= x
(x, y)
0
0
(0,0)
1
1
(1,1)
4
2
(4,2)
9
3
(9,3)
16
4
(16,4)
18
4.24
(18,4.2)
x
Excel Chart
Evaluating
a Square Root Function
Given: f(x) =
Find: f(3)
Solution:
12x – 20
f(3) = 12(3) – 20
= 36 – 20
=
16
=4
Domain of a Square Root Function
Given: f(x) = 3x + 12
Find the Domain of f(x):
Solution:
3x + 12 ≥ 0
3x ≥ -12
X ≥ -4
[-4, ∞)
Application
By 2005, an “hour-long” show on prime time TV was 45.4
min on the average, and the rest was commercials, plugs,
etc. But this amount of “clutter“ was leveling off in recent
years. The amount of non-program “clutter”, in minutes,
was given by:
M(x) = 0.7 x + 12.5
where x is the number of years after 1996.
What was the number of minutes of “clutter” in an hour
program in 2002?
Solution
Solution:
M(x) = 0.7 x + 12.5
x = 2002 – 1996 = 6
M(6) = 0.7 6 + 12.5
~ 0.7(2.45) + 12.5
~ 14.2 (min)
In 2009?
x = 2009 – 1996 = 13
M(13) = 0.7 13 + 12.5
~ 15 (min)
Cube Root and Cube Root Function
3
3
3
a = b,
means b3 = a
8 = 2,
because 23 = 8
-64
= -4
Because (-4)3 = -64
Cube Root Function
f(x) =
3
x
x
y =3 x
(x, y)
-27
-3
(-27,-3)
-8
-2
(-8,-2)
-1
-1
(-1,-1)
0
0
(0,0)
1
1
(1,1)
8
2
(8,2)
27
3
(27,3)
30
3.1
(30,3.1)
Simplifying Radical Expressions
3
-64x3 = 3 (-4x)3 = -4x
4
81 = 4 (3)4 = 3
4 -81 = x has no solution in R,
since there is no x such that x4 = -81
In general
n -a
n -a
has an nth root when n is odd
has no nth root when n is even
7.2 Rational Exponents
What is the meaning of 71/3?
x = 71/3 means
x3 = (71/3)3 = 7
Generally, a1/n is number such that
(a1/n)n = a
Your Turn
Simplify
1.
2.
3.
4.
641/2
(-125)1/3
(6x2y)1/3
(-8)1/3
Solutions
1.
2.
3.
4.
8
-5
3
6x2y
-2
Solve
10002/3
= (10001/3)2 = 102 = 100
163/2
(161/2)3 = 43 = 64
-323/5
-(321/5)3 = -(2)3 = -8
Your Turn
What is the difference
between -323/5 and (-32) 3/5
between -163/4 and (-16) 3/4
Simplify
61/7 · 64/7
+ 4/7) =
32x1/2
--------16x3/4
= 6(1/4
= 2x(1/2 – 3/4) = 2x-1/4
(8.33/4)2/3
65/7
= 8.3(3/4 ∙ 2/3)
= 8.31/2
Simplify
49-1/2
(8/27)-1/3
= 1/(-64)2/3 = 1/((-64)1/3)2 = 1/(-4)2 = 1/16
(52/3)3
= 1/(8/27)1/3 = (27/8)1/3 = 271/3/81/3 = 3/2
(-64)-2/3
= (72)-1/2 = 7-1 = 1/7
= 52/3 ∙ 3 = 52 = 25
(2x1/2)5
25x1/2 · 5 = 32x5/2
7.3 Multiplying & Simplifying
Radical Expressions
Product Rule
a · n b = n ab
or
a1/n · b1/n = (ab)1/n
Note: Factors have same order of root.
n
E.g,
n
25
n
4
=
25 · 4 = 100 = 10
2000 = 400 · 5 = 400 · 5 = 20 5
Simplify Radicals by Factoring
√(80)
3
√(40)
= √(8 · 2 · 5) = √(23 · 2 · 5)
= √(24 · 5) = 4√(5)
3
3
= √(8 · 5) = √(23 · 5)
3
= 2√(5)
√(200x4y2)
= √(5 · 40x4y2) = √(5 · 5 · 8x4y2)
= √(52 · 22 · 2x4y2)
= 5 · 2x2y√(2) = 10x2y√(2)
Simplify Radicals by Factoring
5
√(64x3y7z29)
=
5
√(32 · 2x3y5y2z25z4)
5
= √(25y5z25 · 2x3y2z4)
5
5
= 2yz √(2x3y2z4)
Multiplying & Simplifying
√(15)·√(3)
=
√(45) = √(9·5) = 3√(5)
4
4
3
2
√(8x y )·√(8x5y3)
4
=
√(64x8y5)
=
4
2
2x y√(4y)
4
= √(16·4x8y4y)
Application
Paleontologists use the function
W(x) = 4√(2x)
to estimate the walking speed of a dinosaur, W(x),
in feet per second, where x is the length, in feet, of
the dinosaur’s leg.
What is the walking speed of a dinosaur whose leg
length is 6 feet?
W(x) = 4√(2x)
W(6) =
=
=
=
~
~
4√(2·6)
4√(12)
4√(4·3)
8√(3)
8√(1.7)
14 (ft/sec)
(humans: 4.4 ft/sec walking
22 ft/sec running)
Your Turn
Simplify the radicals
√(2x/3)·√(3/2)
= √((2x/3)(3/2)) = √x
4
4
√(x/3)·√(7/y)
4
4
= √((x/3)(7/y)) = √(7x/3y)
3
√(81x8y6)
3
3
6
2
6
2
2
= √(27·3x x y )= 3x y √(3x2)
3
√((x+y)4)
3
3
3
=√((x+y) (x+y))= (x+y)√(x+y)
7.4 Adding, Subtracting, & Dividing
Adding (radicals with same indices & radicands)
8√(13) + 2√(13)
3
3
3
7√(7) – 6x√(7) + 12√(7)
= √(13) · (8 + 2) = 10√(13)
4
3
3
= √(7) ·(7 – 6x + 12) = (19 – 6x)√(7)
4
4
7√(3x) - 2√(3x) + 2x2√(3x)
4
4
2
2
= √(3x) ·(7 – 2 + 2x ) = (5 + 2x ) √(3x)
Adding
7√(18) + 5√(8)
= 7√(9·2) + 5√(4·2) = 7·3 √(2) + 5·2√(2)
= 21√(2) + 10√(2) = 31√(2)
√(27x) - 8√(12x)
= √(9·3x) - 8√(4·3x) = 3√(3x) – 8·2√(3x)
= √(3x)·(3 – 16) = -13√(3x)
3
√(xy2)
3
3
+ √(8x4y5)
√(xy2)
3
2
3
√(8x3y3xy2)
=
+
= √(xy ) (1 + 2xy)
3
= (1 + 2xy) √(xy2)
=
3
√(xy2)
3
+ 2xy √(xy2)
Dividing Radical Expressions
Recall: (a/b)1/n = (a)1/n/(b)1/n
(x2/25y6)1/2
=(x2)1/2 / (25y6)1/2
=x/5y3
(45xy)1/2/(2·51/2)
= (1/2) ·(45xy/5)1/2 = (1/2) ·(9·5xy/5)1/2
= (1/2) ·3(xy)1/2
= (3/2) ·(xy)1/2
(48x7y)1/3/(6xy-2)1/3
7
-2 1/3
= ((48x y)/6xy ))
= (8x6y3)1/3
= 2x2y
7.5 Rationalizing Denominators
Given: 1
√(3)
Rationalize the denominator—get rid of the
radical in the denominator.
1 √(3) √(3)
=
√(3) √(3)
3
Denominator Containing 2 Terms
Given:
8
3√(2) + 4
Rationalize denominator
Recall: (A + B)(A – B) = A2 – B2
8
3√(2) – 4
8(3√(2) – 4)
=
3√(2) + 4 3√(2) – 4
(3√(2) )2 – (4)2
24 √(2) - 32
8(3 √(2) – 4)
=
18 – 16
12 √(2) - 16
=
2
Your Turn
Rationalize the denominator
2 + √(5)
√(6) - √(3)
2+√(5)
√(6)+√(3)
2√(6)+2√(3)+√(5)√(6)+√(5)√(3)
=
√(6) - √(3) √(6)+√(3)
2√(6) + 2√(3) + √(30) +√(15)
=
3
6–3
7.6 Radical Equations
Application
A basketball player’s hang time is the time in the
air while shooting a basket. It is related to the
vertical height of the jump by the following
formula:
t = √(d) / 2
A Harlem Globetrotter slam-dunked while he
was in the air for 1.16 seconds. How high did
he jump?
Solving Radical Equations
√(x) = 10
(√(x))2 = 102
x = 100
√(2x + 3) = 5
(√(2x + 3) )2 = 52
(2x + 3) = 25
2x = 22
x = 11
Check
√(2x + 3) = 5
√(2(11) + 3) = 5
√(22 + 3) = 5
√(25)
=5
5
=5
?
?
?
yes
Solve
√(x - 3) + 6 = 5
√(x - 3) = -1
(√(x - 3))2 = (-1)2
(x – 3) = 1
x=4
Check:
√(x - 3) + 6 = 5
√(4 - 3) + 6 = 5 ?
√(1) + 6 = 5 ?
1+6
= 5 ? False
Thus, there is no solution to
this equation.
Your Turn
Solve: √(x – 1) + 7 = 2
√(x – 1) = -5
(√(x – 1))2 = (-5)2
x – 1 = 25
x = 26
Check:
√(x – 1) + 7
√(26 – 1) + 7
√(25) + 7
5+7
=2
=2 ?
=2 ?
= 2 ? False
Thus, there is no solution to
this equation.
Your Turn
Solve: x + √(26 – 11x) = 4
√(26 – 11x) = 4 – x
(√(26 – 11x))2 = (4 – x)2
26 – 11x = 16 – 8x + x2
0 = x2 + 3x – 10
x2 + 3x – 10 = 0
(x – 2)(x + 5) = 0
x–2=0
x=2
x+5=0
x = -5
Check -5:
√(26 – 11x) = 4 – x
√(26 – 11(-5)) = 4 – (-5) ?
√(26 + 55) = 4 + 5
?
√(81)
=9
?
9
= 9 True
Check 2:
√(26 – 11x) = 4 – x
√(26 – 11(2)) = 4 – 2 ?
√(4)
=2
?
2
=2
True
Solution: {-5, 2}
Hang Time in Basketball
A basketball player’s hang time is the time spent
in the air when shooting a basket. It is a
function of vertical height of jump.
√(d)
t = ----- where t is hang time in sec and
2 d is vertical distance in feet.
If Michael Wilson of Harlem Globetrotters had
a hang time of 1.16 sec, what was his vertical
jump?
Hang Time
√(d)
t = ----2
2t
= √(d)
2(1.16) = √(d)
2.32 = √(d)
(2.32)2 = (√(d))2
5.38 = d
7.7 Complex Numbers
What kind of number is x = √(-25)?
Imaginary Unit i
x2 = -25?
i = √(-1), i 2 = -1
Example
√(-25) = √((25)(-1)) = √(25)√(-1) = 5i
√(-80) = √((80)(-1)) = √((16 · 5)(-1))
= 4√(5)i
= 4i √(5)
Your Turn
Express the following with i.
1.
2.
3.
4.
√(-49)
√(-21)
√(-125)
-√(-300)
Complex Numbers
Comlex number has a Real part and an
Imaginary part of the form: a + bi
Example
1. 2 + 3i
2. -4 + 5i
3. 5 – 2i
Adding and Subtracting Complex
Numbers
(5 – 11i) + (7 + 4i)
= 5 – 11i + 7 + 4i
= 12 – 7i
(2 + 6i) – (12 – 4i)
= 2 + 6i – 12 + 4i
= -10 + 10i
Multiplying Complex Numbers
4i(3 – 5i)
= 12i – 20i2
= 12i – 20(-1)
= 12 + 12i
(5 + 4i)(6 – 7i)
= 5·6 – 5 ·7i + 4i· 6 – 4 ·7i2
= 30 – 35i + 24i – 28(-1)
= 30 – 11i + 28
= 58 – 11i
Multiplying
1. √(-3) √(-5)
= i√(3) · i√(5)
= i2 √(15)
= -√(15)
2. √(-5) √(-10)
= i√(5) · i√(10)
= i2 √(50)
= -√(50)
= -√(25 · 2)
= -5√(2)
Conjugates and Division
Given: a + bi
Conjugate of a + bi: a – bi
Conjugate of a – bi: a + bi
Why conjugates?
(a + bi)(a – bi)
= (a)2 – (bi)2
= a2 – b2i2
= a2 + b2
(3 + 2i)(3 – 2i) = 9 – (2i)2= 9 – 4(-1) = 13
Multiplying a complex number by its conjugate results in
a real number.
Dividing Complex Numbers
Express 7 + 4i
-------as a + bi
2 – 5i
7 + 4i (7 + 4i) (2 + 5i)
14 + 35i + 8i + 20
-------- = ---------- · ----------- = -----------------------2 – 5 i (2 – 5i) (2 + 5i)
4 + 25
34 – 43i
= ------------29
Your Turn
6 + 2i
-------4 – 3i
6 + 2i (4 + 3i) 24 + 18i + 8i + 6i2
= ---------- · ---------- = ------------------------(4 – 3i) (4 + 3i)
16 + 9
(18 + 26i)
= ------------25
Your Turn
5i – 4
------3i
(5i – 4) -3i
-15i2 + 12i
= --------- · ----- = ------------------3i
-3i
-9i2
15 + 12i 3(5 + 4i) 5 + 4i
= ------------ = ----------- = --------9
9
3
Powers of i
i2
i3
= -1
= (-1)i = -i
i4 = (-1)2 = 1
i5 = (i4)i = i
i6 = (-1)3 = -1
i7 = (i6)i = -i
i8 = (-1)4 = 1
i9 = (i8)i = i
i10 = (-1)5 = -1
Your Turn
Simplify
i17
i17 = i16i = (i2)8i = i
i50
i50
= (i2)25 = (-1)25 = -1
i35
i35
= (i34)i = (i2)17i = (-1)17i = -i
Application
Electrical engineers use the Ohm’s law to relate
the current (I, in amperes), voltage (E, in volts),
and resistence (R, in ohms) in a circuit:
E = IR
Given: I = (4 – 5i) and R = (3 + 7i), what is E?
E = (4 – 5i)(3 + 7i) = 12 + 28i - 15i - 35i2
= 47 + 13i (volts)