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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
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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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
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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PRACTICE: REPRESENTING A FUNCTION (Pt.2)
Express visually.
A
-
C
B
D
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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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.
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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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
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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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) ______________________
_____________________ ______________________ ______________________
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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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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CH.1: PRE-CALC (PART 1)
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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
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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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
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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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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CH.1: PRE-CALC (PART 1)
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PRACTICE: RATIONAL FUNCTIONS #1 PROBLEM: Graph the following function,
A.
C.
B.
D.
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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PRACTICE: RATIONAL FUNCTIONS #2 PROBLEM: Graph the following function,
A.
C.
B.
D.
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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PRACTICE: RATIONAL FUNCTIONS #3 PROBLEM: Graph the following function,
A.
C.
B.
D.
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CH.1: PRE-CALC (PART 1)
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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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CH.1: PRE-CALC (PART 1)
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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,∞)
CALCULUS - CLUTCH
CH.1: PRE-CALC (PART 1)
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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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CH.1: PRE-CALC (PART 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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CH.1: PRE-CALC (PART 1)
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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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CH.1: PRE-CALC (PART 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]
6. Find the range of the following function:
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,∞)
8. Find the range of the following function:
A −∞,−2 ∪ (−2,∞) B −∞, 0 ∪ (0,∞) C (−∞,∞) D (2,∞)
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CH.1: PRE-CALC (PART 1)
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