chapter 7 radical equations. lesson 7.1 operations of functions
TRANSCRIPT
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Chapter 7Radical Equations
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Lesson 7.1Operations of Functions
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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)
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Examples
1. f(x)= x2 – 3x + 1 and g(x) = 4x + 5 find a. (f + g)(x) b. (f – g)(x)
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2. f(x) = x2 + 5x – 1 and g(x) = 3x – 2 finda. (f · g)(x) b. (f/g)(x)
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Composition of Functions
•[f o g](x) = f[g(x)]
▫Plug in the full function of g for x in function f
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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
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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 3. 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)]
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Lesson 7.2 Inverse Functions and Relations
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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 relationsQ = {(1, 2), (3, 4), (5, 6)}S = {(2, 1), (4, 3), (6, 5)}
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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)}
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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 y2. Switch x and y3. Solve for y4. Replace y with f-1(x)5. Graph f(x) and f-1(x) on the same
coordinate plane
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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
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Examples
3. Find the inverse of and then graph the function and its inverse
5
3)(
x
xf
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4. Find the inverse of and then graph the function and its inverse f(x) = 2x - 3
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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
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5. Determine if the functions are inverses
25
1)(
105)(
xxg
xxf
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6. Determine if the functions are inverses
43
1)(
33)(
xxg
xxf
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Lesson 7.3Square Root Functions and Inequalities
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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!!
xy
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Parent Functionsxy 2xy
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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.
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Examples1. Graph. State the domain, range, and x- and y-
intercepts.
43 xy
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2. Graph. State the domain, range, and x- and y- intercepts.
532 xy
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3. Graph. State the domain, range, and x- and y- intercepts
12
3 xy
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Square Root Inequalities
•Follow same steps as an equation to graph but add last step of shading.
•Remember rules for solid and dotted lines!!
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Examples4. Graph 62 xy
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5. Graph 1 xy
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6. Graph
153 xy
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7. Graph
422 xy
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7.4Nth Roots
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Nth Roots
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Symbols and Vocabulary
n #index
Radical Sign
Radicand
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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
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Summary of Nth Roots
n b > 0 b < 0 b = 0
evenone positive root, one negative root
no real roots
One real root = 0
oddone positive root, no negative root
no positive roots, one
negative root
n b n b
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Examples
425.1 x
5 201532.3 yx
82 )2(.2 y
9.4
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Your turn…
681.5 y
6 1830729.7 yx
12)3(.6 x
25.8
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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▫
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Examples Using Absolute Values
8 8.9 x 4 12)1(81.10 a
10100.11 x14)1(64.12 y
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7.5Operations with Radical Expressions
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pg 9
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
If all parts of the radicand are
positive- separate each part so that it
has the nth root
Ex:
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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
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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.
33
2
27
4
27
4
9
32
33
32
3
3
33
2
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Examples
7816.1 qp
10536.3 sr
5
4
.2y
x
7
9
.4n
m
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Simplify all radicals then combine -Remember you can only combine like terms!
1086272125.7
18432583.6
482273122.5
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Foil- then simplify
356224.10
635635.9
323253.8
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Multiply by the conjugate- then simplify
56
523.13
25
24.12
35
31.11
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7.6Rational Exponents
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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 radicalThe numerator becomes the power for the radicand
Ex: = 3
1
83 8
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Examples
5
1
4
1
.2
.1
x
a
5
3
243.4
16.3 4
1
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Generalization for numerator greater than 1
3 28 23 8 22
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: = = = 3
2
8
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Examples
4
3
5
7
5
1
.2
.1
y
xx
5
4
4
9
4
1
.4
.3
r
aa
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1
1.7
9.6
3
81.5
2
1
2
1
4 2
6
8
m
m
z
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2
2.10
16.9
2
32.8
2
1
2
1
3 4
3
4
y
y
x
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7.7Solving Radical Equations and Inequalities
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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.
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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.
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Solve Radical Equations
421 x
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xx 315
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02)15(3 3
1
n
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0262 4
1
y
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Day #2Now let’s look at inequalities
•Steps to Solve a radical inequality1. Identify the values of x for which the root is
defined.2. Solve given inequality by isolating and then
eliminating the radical3.Test values to confirm your solution (use a
table to do this)
4. Graph solution on a number line. (Remember open and closed dots)
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6442 x
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5122 x
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4244 x
pg 14