calculus - clutch ch.1: pre-calc (part...

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CALCULUS - CLUTCH

CH.1: PRE-CALC (PART 1)

FUNCTIONS: INTRO & REPRESENTATION

● A function is the relationship between ________________ and ________________ .

- There are FOUR ways to represent a function:

● All functions need a(n) __________________ & __________________ variable:

- The Independent Variable is related to the function’s ________________.

- The Dependent Variable is related to the function’s ________________.

EXAMPLE 1: According to Mike’s doctor, he will be growing

two inches every year. He is currently 5 ft tall. Numerically

show his height over the next 4 years.

EXAMPLE 2: Express f(x) = x2 + 1 numerically and visually.

Verbally

Visually Algebraically

Numerically

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PRACTICE: REPRESENTING A FUNCTION (Pt.1) 1. Brian has been working out, and is losing 4 pounds every week. He is currently 160 pounds. Express his weight

numerically over the next 4 weeks.

A

x y 0 160

1 164

2 168

3 172

4 176

C

x y 0 164

1 168

2 172

3 176

4 180

B

x y 0 160

1 156

2 152

3 148

4 144

D

x y 0 160

1 40

2 10

3 2.5

4 0

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PRACTICE: REPRESENTING A FUNCTION (Pt.2)

Express visually.

A

-

C

B

D

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WORD PROBLEMS AS FUNCTIONS ●Identify the ______________ used in the problem. EXAMPLE 1: A rectangle has an area of 81 m2. Express the perimeter of the rectangle as a function of the length.

EXAMPLE 2: An open rectangular bin with a volume of 8 m3 has a square base. Express the surface area of the bin as a

function of the length of the base.

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PRACTICE: Word Problems as Functions 1.Maria is planning on renting a car for 7 days to visit Disney World. She is landing in Miami and was looking at two

rental companies. Company A charges $50 a day and 15 cents per mile. Company B charges $60 and charges 10

cents per mile. She plans to drive 1000 miles total. What company has the lower cost?

2.Larry has to make an open top box with a sheet of cardboard with dimension 30 by 12. He is to cut out equal

squares of side x at each corner, then fold up each side. Express the volume V of the box as a function of x.

A ( ) ( )( )( ) C C. ( ) ( )( )( ) B ( ) ( )( )( ) D D. ( ) ( )( )( )

A Company A

B Company B

C Both are Equal

D None of the Above

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TYPES OF FUNCTIONS ● We will refer to this as your _______________ of Functions. These are the 6 basic functions that will be mainly used:

The Linear Function The Parabola The Cubic Function

The Absolute Value The Square-Root Function The Cube-Root Function

EXAMPLE 1: Determine what parent function each graph belongs with:

(a) ______________________

(b) ______________________

(c) ______________________

_____________________ ______________________ ______________________

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PRACTICE: TYPES OF FUNCTIONS

PROBLEM: Which parent function does this graph go with?

1.

2.

A

B | |

C

D √

A

B √

C

D √

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EXTRA: ASYMPTOTES

● Asymptotes are equations, not just numbers. Ex: _______________ & _______________

VERTICAL ASYMTOTES (VA)

● They exist wherever the (Numerator/ Denominator) of rational functions is zero. Then, we say the fraction is __________

● Polynomial functions do NOT have vertical asymptotes.

HORIZONTAL ASYMTOTES (HA)

WHEN EXAMPLE ANSWER

Top exponent is greater

Bottom exponent is greater

The exponents are the same.

OBLIQUE/SLANT ASYMTOTES

● Only occur when the degree of the numerator is greater than the denominator by exactly (one / two/ three)degree(s).

●To find these asymptotes, we use ____________ _______________ .

EXAMPLE 1: Find all the asymptotes of

EXAMPLE 2: Find all the asymptotes of

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PRACTICE: ASYMPTOTES

PROBLEM: Find the Vertical Asymptotes,

1.

.

2.

.

PROBLEM: Find the Horizontal Asymptotes,

3.

.

4

A

B

C

D No asymptote

A

B

C

D No asymptote

A

B

C

D No asymptote

A

B

C

D No asymptote

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TYPES OF FUNCTIONS: RATIONAL ●The Rational Function: ●A rational function is the ratio of two __________________ It can be express as: __________________ .

● Given that ______ and __________ are polynomials and ____________.

● Other than horizontal & vertical asymptotes, we should look out for _______________ asymptotes.

EXAMPLE 1: Graph

EXAMPLE 2: Graph

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PRACTICE: RATIONAL FUNCTIONS #1 PROBLEM: Graph the following function,

A.

C.

B.

D.

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A.

C.

B.

D.

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COMPUTING DOMAIN ● Domain is the set of inputs of X. ● Which values of X are not allowed? POLYNOMIALS ● They (do / do not) have restrictions on their domain.

A) 𝑓 𝑥 = 𝑥$ − 1

B) 𝑓 𝑥 = 𝑥' − 4𝑥) + 2

RADICALS ● Inside of radical (radicand) has to be (negative / positive / zero).

A) 𝑓 𝑥 = 𝑥 − 1

B) 𝑓 𝑥 = 𝑥 + 5-

RATIONAL FUNCTIONS ● (Numerator / Denominator) can’t equal (≠) zero. The fraction is _________________ .

A) 𝑓 𝑥 = '12$

B) 𝑓 𝑥 = 31453

OTHER ● We must find the (union / intersection) of domains when dealing with multiple functions.

A) 𝑓 𝑥 = 12$153

B) 𝑓 𝑥 = $2112'

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PRACTICE: COMPUTING DOMAIN (Pt 1) PROBLEM: Find the domain for each of the following functions: 1. 𝒇 𝒙 = 𝒙𝟐 + 𝟐𝒙 − 𝟏

2. 𝒇 𝒙 = 𝒙 − 𝟏 + 𝟓

3. 𝒇 𝒙 = 𝒙5𝟏𝒙𝟐5𝟕𝒙5𝟏𝟎

A (−1,∞) B (−∞,−1) C (−∞,∞) D (0,∞)

A [−1,∞) B [1,∞) C [5,∞) D (−∞,∞)

A (-5,-2) B (−∞,−5) ∪ (−2,∞) C −∞,−5 ∪ (−5,−2) ∪ (−2,∞) D (−2,∞)

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PROBLEM: Find the domain for each of the following functions:

4. 𝒇 𝒙 = 𝟐𝒙5𝟐

5. 𝒇 𝒙 = 𝟏𝒙2𝟑

6. 𝒇 𝒙 = 𝒙5𝟓𝒙2𝟏

A (−∞, 2) ∪ (2,∞) B (−∞,−2) ∪ (−2,∞) C (−2,2) D (−∞,∞)

A [3,∞) B (3,∞) C (−∞,−3) D (−∞,∞)

A [−5,∞) B [−5,1) ∪ (1,∞) C (−1,∞) D −5,1 ∪ [−1,∞)

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DOMAIN AND RANGE ● The Domain is the (input / output) and the Range is the(input / output). ● The _________________ is the set of inputs of X. ● The _________________ is the set of all values (Y) that the function takes when x is inputted. ● Polynomials don’t have restrictions in their (Domain / Range).

Domain: _________________ Domain: _________________

Range: _________________ Range: _________________

EXAMPLE 1 EXAMPLE 2 EXAMPLE 3

𝒚 = −(𝒙 − 𝟏)𝟐 𝒚 = 𝒙 + 𝟐 𝒚 =𝟏𝒙

Domain: Domain: Domain:

Range Range Range

Vertical asymptotes affect:

Horizontal asymptotes affect:

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PRACTICE: DOMAIN AND RANGE PROBLEM:ComputetheDomain&Range.

1.Giventhefollowingsetofcoordinates,findthedomain:

{ −2,2 , −1,3 , 0,4 , 1,5 , (2,6)}

A {−2, −1,0,1,2} B {2,3,4,5,6} C {−2, −1,3,4,5,6} D {∅}

2.Giventhefollowingsetofcoordinates,findtherange:

{(−2,2)(−1,3)(0,4)(1,5)(2,6)}

A {2,3,4,5,6} B {−2, −1,0,1,2} C {−2, −1,3,4,5,6} D {∅}

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PROBLEM: Compute the Domain & Range.

3. Find the domain of the following function:

A [1,∞) B [2,∞) C (−∞, 2) D (−∞, 1]

4. Find the range of the following function:

A [2,∞) B (−∞, 1] C (−∞, 2) D [1,∞)

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PROBLEM: Compute the Domain & Range.

5. Find the domain of the following function:

A [−1,∞) B [−2,∞) C (−∞,∞) D (−∞, 1]


A (−∞, 1) B (−1,∞) C (−∞,∞) D [−1,∞)

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PROBLEM: Compute the Domain & Range. 7. Find the domain of the following function:

A (−∞,∞) B −∞, 0 ∪ (0,∞) C −∞,−2 ∪ (−2,∞) D (2,∞)


A −∞,−2 ∪ (−2,∞) B −∞, 0 ∪ (0,∞) C (−∞,∞) D (2,∞)

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calculus - clutch ch.1: pre-calc (part...

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