factoring, exponents, and functions notes.notebook...jan 29, 2015 · factoring, exponents, and...

Factoring, Exponents, and Functions Notes.notebook

1

January 29, 2015

This material is made freely available at www.njctl.org and is intended for the noncommercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.

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New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative Algebra II

Fundamental Skills of

Algebra

www.njctl.org

20130724

Table of Contents

Factoring

Exponents

Functions

click on the topic to go to that section

Factoring

Return to Table of Contents

Factoring

Goals and ObjectivesStudents will be able to factor complex expressions and solve equations using factoring.

Factoring

Why do we need this?The more we can simplify a problem, the easier it is to solve. Factoring allows us to break up expressions into smaller parts and, much of the time, simplify our

strategies to solve them.


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January 29, 2015

Factoring

Multiply the following:

3x(2x2 + 5) (x + 3)(x 9) (2x 1)(3x 4)

Answer

Factoring

Factoring is undoing what you just did. It breaks up expressions into parts and pieces.

Factoring

Factor out the GCF (greatest common factor).

8x + 32y 5a2b 10ab 12a3b2 + 4ab2

Answer

8x + 32y8(x + 4y)

*If students struggle with this have them write out each term as lowest common factors. For example:

8x+32y=2*2*2*x+2*2*2*2*2y OR 5a2b10ab=5*a*a*b2*5*a*b

Students can then match the pieces each term has in common.

Factoring

You can easily check your answer by distributing the GCF back to all of your terms.

Factor then check your answer:

3x2y2 + 12xy2 + 6y2 6m4n3 + 18m3n2 36m2n

Answer

3x2y2 + 12xy2 + 6y23y2(x2 + 4x + 2)

1 What is the GCF of the following expression? 24a3b3c 6ab3c

A 6a3b3c B 6abc C ab3c D 6ab3c E a3b3c

Factoring

Answer

2 Factor:

A 3m5n2(4 n 5m)

B m5n2(12 3n 15m)

C 3m6n3(4 5n)

D 3m4n2(4mn 5m2)

E 3m4n2(4 mn 5m2)

Factoring

12m4n2 3m5n3 15m6n2

Answer


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January 29, 2015

3 Factor out the GCF:

A x2y2z3(x2 xy + y2)

B xyz(x2 xy + y2)

C x3y2z3(x y y2)

D x2y3z3(x2 x y)

E x2y2z3(x xy + y2)

x4y2z3 x3y3z3 + x2y4z3Factoring

Answer

4 Factor the following:

A 4x3y2(2x + 3y)

B 4x3y2(2x 3y)

C 8x3y2(x + 4y)

D 8x3y2(x 4y)

E 8x4y2(1 4y)

8x4y2 12x3y3Factoring

Answer

5 Factor out the GCF: 15m3n 25m2 15mn3

A 15m(mn 10m n3)

B 5m(3m2n 5m 3n3)

C 5mn(3m2 5m 3n2)

D 5mn(3m2 5m 3n)

E 15mn(mn 10m n3)

Factoring

Answer

6 Factor:

A m3n3(2m 2)

B 2mn(m3n + m)

C 2mn(m3n m)

D 2m3n3(m2 + 1)

E 2m3n3(m2 1)

Factoring

2m4n3 2m3n3

Answer

7 Factor out the GCF:

A 14p3q7(1p 14)

B 14p2q6(pq 14)

C 14p2q6(pq 2)

D p3q6(14q 28)

E 14pq(p2q6 2pq5)

Factoring

14p3q7 28p2q6

Answer

Factoring

Factoring quadratics of the form x2 + bx + c.

Try: x2 x 6

Answer


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January 29, 2015

Factoring

Here are some tips to help you factor:

Read the expression backwards:

x2 x 6

"factors of 6 that subtract to 1"

x2 + 6x + 8

"factors of 8 that add up to 6"

Factoring

The signs in front of c will also help you factor:

x2 + bx + c ⇒ (x + m)(x + n)

x2 bx + c ⇒ (x m)(x n)

x2 + bx c

x2 bx c

If the sign in front of c is +: If the sign in front of c is :

(x m)(x + n)⇒

⇒

same signs, both +

same signs, both

different signs

different signs

Factoring

Here is the last tip...

Use a factor tree to help you find the factors of c. If none of the pairs add or subtract to the middle term, the quadratic is not factorable.

Factor trees: 12

>

1 122 63 4

24

>

1 242 123 84 6

48

>

1 482 243 164 126 8

72

>

1 722 363 244 186 128 9

Factoring

x2 + 6x + 8

Read it backwards: "factors of 8 that add up to 6."

Use the pattern from c: "same signs, both +."

Now factor it!

Answer

Factoring

Use the tips to help you factor the following quadratics.

x2 5x 24 a2 13a + 30 m2 + 4m 35

Answer

x2 5x 24(x 8)(x + 3)

Factoring

Try...x2 8xy + 16y2

Answer


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January 29, 2015

8 Factor the trinomial:

A (x 7y)(x + 6y)

B (x + 7y)(x 6y)

C (x 42y)(x + y)

D (x + 42y)(x y)

E Solution not shown

Factoring

x2 xy 42y2

Answer

9 Factor:

A (x 12)(x + 2)

B (x + 12)(x 2)

C (x 6)(x +4)

D (x 6)(x 4)


Factoring

x2 10x + 24

Answer

10 Factor the quadratic:

A (x + 5)(x + 1)

B (x 4)(x 1)

C (x + 2)(x + 3)

D (x + 1)(x 4)


x2 + 5x + 4Factoring

Answer E

Answer is (x + 4)(x + 1)

11 Factor:

A (x 12)(x 6)

B (x 6)(x 3)

C (x 6)(x + 3)

D (x 9)(x + 2)


x2 3x 18

Factoring

Answer

12 Factor:

A (x 5)(x 5)

B (x 5)(x + 5)

C (x + 15)(x + 10)

D (x 15)(x 10)


Factoring

x2 + 10x + 25

Answer

FactoringFactoring quadratics of the form ax2 + bx + c.

3x2 + 17x + 10

With just a few extra steps, you can factor quadratics with a leading coefficient (a) just like the previous ones.

Mulitiply a and c ⇒ x2 + 17x + 30Factor the result ⇒ (x + 15)(x + 2)

Divide the numbers by a ⇒ (x + 15)(x + 2) 3 3

Reduce any fractions ⇒ (x + 5)(x + 2)3

Move any remaining denominators ⇒ (x + 5)(3x + 2)

The answer is (x + 5)(3x + 2) Teacher N

otes There are other methods you can

use to factor these. This is one of the most efficient ways. Students catch on quickly. The most common mistake is forgetting to divide and/or reduce.


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January 29, 2015

Factoring

Try this one. Make sure you reduce the fractions where possible.

10x2 31x + 15

Answer

Factoring

Here is another one to practice on...

3x2 + 5x 2

Answer

Factoring

Try... 4x2 + 4x +1

Answer

13 Factor:

A (6x 5)(x 6) B (6x 1)(x 6) C (3x 2)(2x + 3) D (3x + 2)(2x 3) E Solution not shown

Factoring

6x2 5x 6

Answer

14 Factor the following:

A (2x 1)(5x + 3) B (2x + 1)(5x + 3) C (10x 1)(x + 3) D (10x 1)(x 3) E Solution not shown

10x2 11x + 3

Factoring

Answer

15 Which is a factor of ?

A (4x + 5)B (2x + 3)C (12x + 5)D (3x + 5)E (4x + 2)

12x2 + 23x + 10Factoring

Answer


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January 29, 2015

16 Which is a factor of the quadratic?

A (x 7y)B (2x 8y)C (2x 6y)D (x 8y)E (2x y)

Factoring

2x2 23xy + 56y2

Answer

17 Which of the following is a factor of ?

A (2x 3y)B (5x 3y)C (3x 2y)D (3x + 5y)E (2x 5y)

6x2 19xy + 15y2Factoring

Answer A

Solution is (2x 3y)(3x 5y)

18 Factor:

A (x 4y)(4x + 2y)B (x 8y)(4x + y)C (2x + 4y)(2x + 2y)D (4x + y)(x + 8y)E Solution not shown

Factoring

Answer

Factoring

Multiply:

(2x + 3)(2x 3) (x + 3)(x2 3x + 9)

Answer

(2x + 3)(2x 3)4x2 9

To factor the difference of squares, the difference of cubes and the sum of cubes, use the following formulas:

Factoring

a2 b2 ⇒ (a b)(a + b)

a3 b3 ⇒ (a b)(a2 + ab + b2)

a3 + b3 ⇒ (a + b)(a2 ab + b2)

Try...

Factoring

4p2 q2 16m2 1 64p3 + y3

a2 b2 ⇒ (a b)(a + b)a3 b3 ⇒ (a b)(a2 + ab + b2)a3 + b3 ⇒ (a + b)(a2 ab + b2)

Answer


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January 29, 2015

Factoring

a2 b2 ⇒ (a b)(a + b)a3 b3 ⇒ (a b)(a2 + ab + b2)a3 + b3 ⇒ (a + b)(a2 ab + b2)Factor...

25x2 81y2 x3y3 + 1 8m3 125n3

Answer

19 Factor:

A (11m 10n)(11m + 10m) B (121m n)(m + 100n) C (11m n)(11m + 100n) D Not factorable E Solution not shown

121m2 + 100n2Factoring

Answer

20 Factor:

A (3m 8np)(3m + 8np) B (3m + 4np)(3m2 12mnp + 4n2p2) C (3m + 4np)(9m2 12mnp + 16n2p2) D Not factorable E Solution not shown

Factoring

27m3 + 64n3p3

Answer

21 Factor:

A (a 5b)(a + 5b) B (a 5b)(a2 + 5ab + 5b2) C (a + 5b)(a2 5ab + 25b2) D Not FactorableE Solution not shown

Factoring

a3 125b3

Answer

22 Factor:

A (m + n)(m n) B (m + n)(m + n) C (m n)(m n) D Not factorableE Solution not shown

Factoring

m2 n2

Answer

23 Factor:

A (d 1)(d + 1) B (d 1)(d2 + d + 1) C (d + 1)(d2 d + 1) D Not factorableE Solution not shown

Factoring

d3 1

Answer


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January 29, 2015

24 Factor:

A (3m + 2n)(3m 2n) B (3m + 2n)(9m2 6mn + 4n2) C (3m 2n)(9m2 + 6mn + 4n2) D Not factorableE Solution not shown

Factoring

27m3 + 4n3

Answer

25 Factor:

A (16x 2y)(16x + 2y) B (6x 2y)(6x2 + 12xy + 2y2) C (6x 2y)(36x2 + 12xy + 4y2) D Not factorableE Solution not shown

Factoring

216x3 8y3

Answer

Factoring

Factoring by Grouping.

What happens when there are 4 terms?

4ap 4a + 3xp 3x

Answer

Factoring

Try... xy + 4x 3y 12

Answer

Factoring

Two more...pq + 4p + 3q + 12 mn pm qn + qp

Answer

pq + 4p + 3q + 12p(q + 4) + 3(q + 4)(p + 3)(q + 4)

26 Factor by grouping:

A (r + 4)(s 7) B (r 7)(s + 4) C (rs 7)(rs + 4) D Not factorableE Solution not shown

Factoring

rs 7r + 4s 28

Answer


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January 29, 2015

27 Factor:

A (n 3)(m + 4n) B (n 3)(m 4n) C (n + 4)(m n) D Not factorableE Solution not shown

Factoring

mn + 3m 4n2 12n

Answer

28 Factor:

A (3k 2)(g + 6) B (3g + 2)(k 6) C (3k 2)(g + 6) D Not factorableE Solution not shown

Factoring

3gk 18g + 2k 12

Answer

29 Factor by grouping:

A (m 4)(2p 7) B (m + 7)(2p + 4) C (m 4)(2p + 7) D Not factorableE Solution not shown

Factoring

2mp 8p 7m + 28

Answer

30 Factor by grouping

A (x + 4)(2y + 3) B (x + 4)(2y 3) C (x 3)(2y + 4) D Not factorableE Solution not shown

2xy + 3x + 8y 12Factoring

Answer

31 Factor:

A (3m 5)(p + 2n) B (3m + 5)(p 2n) C (3m 2n)(p + 5) D Not factorableE Solution not shown

Factoring

3mp + 15m 2np 10n

Answer

Now, let's combine all of the situations. In any factoring problem, factor out the GCF first.

Factoring

Factor these completely...

2x3 22x2 + 48x 3m3n + 3m2n 18mn

Answer

2x3 22x2 + 48x2x(x2 11x + 24)2x(x 8)(x 3)


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January 29, 2015

Factoring

Factor completely...

4x3 32y3 54a4 + 2ab3

Answer 4x3 32y3

4(x3 8y3)4(x 2y)(x2 + 2xy + 4y2)

32 Factor completely:

A 3mn(2m 1)(m 3) B 3mn(2m + 1)(m 3) C 3n(2m 1)(m2 + 3m + 9) D Not factorableE Solution not shown

Factoring

6m3n + 21m2n 9mn

Answer


A 3(2p + 5)(4p3 10p + 25) B 3p(16p2 + 25) C 3p(4p 5)(4p 5) D Not factorableE Solution not shown

Factoring

48p3 + 75p

Answer


A (4p2 3)(m 4) B 4p2(pm 4)(pm 3) C 4p2m(m 4)(p + 3) D Not factorableE Solution not shown

Factoring

4p3m 12p3 16mp2 + 48p2

Answer


A 2xy(9x 1) B 2xy(3x 1)(3x + 1) C 2y(9x2 x) D Not factorableE Solution not shown

Factoring

18x3y 2xy

Answer


A 8ab(a2 + 4) B 4ab(2a2 + a + 3)C 4ab(2a + 3)(a 1)D Not factorableE Solution not shown

Factoring

8a3b 4a2b + 12ab

Answer


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January 29, 2015


A 2b(4b 1)(3b + 2) B 4b(6b2 + 6b 1) C 2b(4b + 1)(3b 2) D Not factorableE Solution not shown

24b3 + 10b2 4b

Factoring

Answer

Factoring is often use to solve equations that are in polynomial form.

Factoring

Steps: 1) Move all terms to one side of the equation. (the other side becomes zero)

2) Factor the resulting polynomial.3) Set each factor equal to zero.4) Solve each equation. 5) Write the answers clearly.

Factoring

Solve the equation by factoring:

x2 = 9x 18

Answer

One more...

Factoring

6x3 + 10x2 = 4x

Answer

Another example...

Factoring

6m2 = 9m 24m3

Answer

38 Which of the following are solutions to the equation?

A 0B 1C 3/2D 3/2E 4F 4G 4/3H 4/3

Factoring

16x3 36x = 0

Answer


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January 29, 2015

39 Find all of the solutions to:

A 0B 1C 1D 3E 3F 4G 4H 12

Factoring

3m3 + 3m2 = 36m

Answer

40 Solve for p:

A 0B 1/2C 1/2D 2/5E 2/5F 4/3G 5/2H 5/2

Factoring

29p2 + 10p = 10p3

Answer

41 Find the values for n:

A 0B 1/2C 1/2D 1/3E 2/3F 2/3G 4H 4

Factoring

18n4 + 48n2 = 84n3

Answer

42 Find x:

A 0B 1/4C 1/4D 1/3E 1/3F 1/2G 1/2H 3

Factoring

6x4 = 5x3 x2

Answer

43 Solve by factoring:

A 0B 1/4C 1/4D 1/2E 1/2F 1G 2H 2

Factoring

p2 + p = 2p3

Answer

Exponents

Return to Table of Contents


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January 29, 2015

Exponents

Goals and ObjectivesStudents will be able to simplify complex expressions containing exponents.

Exponents

Why do we need this?Exponents allow us to condense bigger

expressions into smaller ones. Combining all properties of powers

together, we can easily take a complicated expression and make it

simpler.

Exponents

Rules for working with exponents:

Exponents

Multiplying powers of the same base:

(x4y3)(x3y)

Teacher N

otes Have students think about what the expression

means and then use the rule as a short cut.

For example: (x4y3)(x3y) = x x x x y y y x x x y = x7y4

The rules are a quick way to get an answer.

Exponents

(3a3b2)(2a4b3)

Simplify:

(4p2q4n)(3p3q3n)

Answer

Work out:

Exponents

xy3 x5y4. (3x2y3)(2x3y)

Answer


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January 29, 2015

44 Simplify:

A m4n3p2 B m5n4p3 C mnp9 D Solution not shown

(m4np)(mn3p2)Exponents

Answer

45 Simplify:

A x4y5 B 7x3y5 C 12x3y4 D Solution not shown

Exponents

(3x3y)(4xy4)

Answer

D

(3x3y)(4xy4)12x4y5

46 Work out:

A 6p2q4 B 6p4q7 C 8p4q12 D Solution not shown

Exponents

2p2q3 4p2q4.

Answer

D

2p2q3 4p2q48p4q7

47 Simplify:

A 50m6q8 B 15m6q8 C 50m8q15 D Solution not shown

Exponents

.5m2q3 10m4q5

Answer

48 Simplify:

A a4b11 B 36a5b11 C 36a4b30 D Solution not shown

(6a4b5)(6ab6)

Exponents

Answer

Exponents

Dividing powers with the same base:

Teacher N

otes

Have students think about what the expression means and then use the rule as a short cut.

For example:



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January 29, 2015

Exponents

Simplify:

Answer

Exponents

Try...

Answer

49 Divide:

A

B

C

D Solutions not shown

Exponents

Answer

50 Simplify:

A

B

C

D Solution not shown

Exponents

Answer

D

51 Work out:

A

B

C


Exponents

Answer

52 Divide:

A

B

C


Exponents

Answer


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January 29, 2015

53 Simplify:

A

B

C


Exponents

Answer

D

Exponents

Power to a power:

Teacher N

otes

Have students think about what the expression means and then use the rule as a short cut.

For example:


Exponents

Simplify:

Answer

Try:

Exponents

Answer

54 Work out:

A

B

C


Exponents

Answer

55 Work out:

A

B

C


Exponents

Answer


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January 29, 2015

56 Simplify:

A

B

C


Exponents

Answer

57 Simplify:

A

B

C


Exponents

Answer

D

58 Simplify:

A

B

C


Exponents

Answer

Negative and zero exponents:Exponents

Why is this? Work out the following:

Teacher N

otes

Exponents

Sometimes it is more appropriate to leave answers with positive exponents, and other times, it is better to leave answers without fractions. You need to be able to translate expressions into

either form.

Write with positive exponents: Write without a fraction:

Answer

Exponents

Simplify and write the answer in both forms.

Answer


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January 29, 2015

Exponents

Simplify and write the answer in both forms.

Answer

Exponents

Simplify:

Answer

Exponents

Write the answer with positive exponents.

Answer

59 Simplify and leave the answer with positive exponents:

A

B

C


Exponents

Answer

60 Simplify. The answer may be in either form.

A

B

C


Exponents

Answer

61 Write with positive exponents:

A

B

C


Exponents

Answer

D


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January 29, 2015

62 Simplify and write with positive exponents:

A

B

C


Exponents

Answer

D

63 Simplify. Write the answer with positive exponents.

A

B

C


Exponents

Answer

64 Simplify. Write the answer without a fraction.

A

B

C


Exponents

Answer

CombinationsExponents

Usually, there are multiple rules needed to simplify problems with exponents. Try this one. Leave your answers with positive exponents.

Answer

Exponents

When fractions are to a negative power, a short cut is to flip the fraction and make the exponent positive.

Try...

Answer

Exponents

Two more examples. Leave your answers with positive exponents.

Answer


21

January 29, 2015


A

B

C


Exponents

Answer

D

66 Simplify. Answer can be in either form.

A

B

C


Exponents

Answer


A

B

C


Exponents

Answer

68 Simplify and write without a fraction:

A

B

C


Exponents

Answer

Operations with Functions

Return toTable ofContents

Goals and ObjectivesStudents will be able to manipulate multiple functions algebraically and simplify resulting

functions.


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January 29, 2015

Why do we need this?In this unit, we will graphically explored transformations of functions. Sometimes, data is more complex and requires more than one representative function. Algebraically, manipulating functions allows us to combine different functions together and results in

many more options for real life situations.

Here are the properties of combining functions:

Adding functions:

Subtracting functions:

Multiplying functions:

Dividing functions:


Answer

Given: and

Find: Simplify your

answers as much as possible.

What happens to the domain?

Operations with Fractions

Answer

Given: and

Find:


Answer

69 Given and , find

A

B

C

D Answer

70 Given

A

B

C

D

and , find h(x) if

Answer


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January 29, 2015

71 Given and , find

A

B

C

D

Answer

72 Given and , find

A

B

C

D Answer

Function Notation

For example: y = 3x + 2 becomes f(x) = 3x + 2

*f(x) = 3x + 2 is still a line with a slope of 3 and a yintercept of 2.

Mathematically, nothing changes. Yet, when a relation is a function, instead of writing y mathematicians use f(x), read "f of x."

So why change the notation?

1) It lets the mathematician know the relation is a function.

2) If a second function is used, such as g(x) = 4x, the reader will be able to distinguish between the different functions.

3) The notation makes evaluating at a value of x easier to read.

Function Notation

Evaluating a Function To Evaluate in y = Form:

Find the value of y = 2x + 1when x = 3

y = 2x + 1y = 2(3) + 1

y = 7

When x is 3, y = 7

To Evaluate in Function Notation:

Given f(x) = 2x + 1 find f(3)

f(3) = 2(3) + 1f(3) = 7

"f of 3 is 7"

Similar methods are used to solve but function notation makes asking and answering questions more concise.

Evaluate the function at each given value:

Answer


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January 29, 2015

Evaluate the functions at each given value:

Answer

Find:

Evaluate the functions You can also substitute expressions in for values of x.

Answer

73 Given and Find the value of .

Answer


Answer


Answer


Answer


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January 29, 2015

77 Given and Find:

A

B

C

D

E

Answer

78 Given and Find:

A

B

C

D

E

Answer

79 Given and Find:

A

B

C

D

E Answer

Given and , find:

a)

b)

c)

d)

You may also be asked to evaluate combined functions when given specific values for x.

Combined Functions

Answer

80 Given and , find

A

B

C

D

6

4

12

10

Answer

81 Given and , find

A

B

C

D

1728

864

864

1288

Answer


26

January 29, 2015

82 Given and , find

A

B

C

D

undefined

Answer

Expressions may also be used to create more complex functions.

If and , create .

Leave your answer in terms of x.

Answer

If and , create .Leave your answer in terms of x.

Answer

If and , create .

Leave your answer in terms of x.

Answer

83 If and , create .

Is this equivalent to ?

Yes

No Answer



Yes

No

Answer


27

January 29, 2015



Yes

No

Answer

Composite Functions


or

Goals and ObjectivesStudents will be able to recognize function notation and correctly unite two or more functions together to

create a new function.

Why do we need this?On many occasions, multiple situations happen to something before it obtains a final result. For example, you take extra food off of your plates before you put them in the dishwasher. Or, to wrap a present you must first put it in the box, then apply the wrapping paper, and finally tie the bow. These are multiple

functions that go together to obtain a desired result.

Composite functions exist when one function is "nested" in the other function.

There are 2 ways of writing a composite function:

Each form is read "f of g of x" and both mean the same thing.

or

Composite FunctionsTo simplify composite functions, substitute one function into the other in place of "x" and simplify. Work from the inside out.

Given:

Find: Find:

Composite Functions

Answer f(g(x)) = f(g(x))


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January 29, 2015

Given:

Find: Find:

Composite Functions

Answer

To simplify composite functions with numerical values, substitute the number into the "inner" function, simplify, and then substitute that value in for the variable in the "outer" function.

Given:

Find:

Composite Functions

Find:

Answer

86 If and , find the value of

A B C D

Answer

87 Find if

A B C D

Answer

88 Find

A B C D A

nswer

89 Find

A

B

C

D

Answer


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January 29, 2015

90 Find if

A

B

C

D

Answer

91 Find if and

A

B

C

D

62

88

82

19

Answer

Inverse Functions


Goals and ObjectivesStudents will be able to recognize and find an inverse function:

a) using coordinates,b) graphically and c) algebraically.

Why do we need this?Sometimes, it is important to look at a problem from

the inside out. Addition undoes subtraction. Multiplication undoes division. In order to look deeply into different problems, we must try to see things from the inside out. Inverse functions undo original functions.

Def: An inverse function is a function that undoes another function. The notation for the inverse of is .

You can prove that a function is an inverse of another using the following relationship:

Read, "the inverse of f of x."

Example: Prove that f(x) and g(x) are inverse functions.

Inverse Function

Answer


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You can also prove that two functions are inverses by GRAPHING. The graph of an inverse is the reflection of the function over y = x.

The following functions are inverses of each other. Look at their graphs and make a conjecture about the x and y values of inverse functions.

Inverse Function

Teacher N

otes The inverse of a function is the reflection over y = x. As you

observed, this will result in the switching of x and y values.

Examples:

a) Find the inverse of: b) Find the inverse of:

f(x)= {(1, 2), (3, 5), (7, 6)} X Y

3 2

4 4

5 5

6 7

Inverse Function

Answer

92 What is the inverse of {(1, 4), (5, 3), (2, 1)}?

A {(4,1), (3,5), (2,1)}

B {(1,4), (5,3), (2,1)}

C {(4,1), (3,5),(1,2)}

D {(4,1), (3,5), (1,2 )} Answer

93 If the inverse of a function is {(1, 0), (3, 3), (4, 5)}, what was the original function?

A {(0,1), (3,3), (5,4)}

B {(1,4), (5,3), (2,1)}

C {(0,1), (3,3),(4,5)}

D {(0,1), (3,3), (4,5)} Answer

94 What is the inverse of:A B

C D

Answer

95 Will the inverse of the following points be a function? Why or why not?

Yes

No

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Draw the inverse of the given function.

Is the inverse a function? Why or why not?

Answer

InverseDraw the inverse of the given function.

Is the inverse a function? Why or why not?

Inverse

Answer

Just like the Vertical Line Test, there is a simple way to determine if a function's inverse is also a function, just from looking at its graph.Def: The Horizontal Line Test is used to determine if a function has an inverse. If a horizontal line crosses the function more than once, it WILL NOT have an inverse.

Move this line to check:

Answer

Technically, the horizontal line test is checking to see if the inverse is a function. Every function has an inverse, but it isn't necessarily a function.

This one does not pass!

Horizontal Line TestExample: Will the inverse of the given function be a function?

10 8 6 4 2 0 2 4 6 8 10

1098765432

12345678910

x

y

Move this line to check:

Answer

Horizontal Line Test

96 Which graph is the inverse of the function at right?

A B C D

A B C D

Answer


A B

C D Answer


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C D B A Answer

99 Will the inverse of be a function?

Yes

No

Answer

Knowing that the inverse of a function switches x and y values, we can take this concept further when given an equation.

Given: Switch x and y. Solve for y: Inverse function:

Finding the Inverse of a Function Algebraically

InversePractice: Find the inverse of the following functions.

Answer

Answer

InversePractice: Find the inverse of the following function.

100 Which of the following choices is the inverse of

A

B

C

D

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101 Which of the following is the inverse of

A

B

C

D

Answer

102 Find the inverse of

A

B

C

D

Answer

103 Find the value of

A

B

C

D

0

1

2

1

Answer

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