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FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008
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FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Jan 04, 2016

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Page 1: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

FUNCTIONS & GRAPHS2.1

JMerrill, 2006Revised 2008

Page 2: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Definitions What is domain? Domain: the set of input values (x-

coordinates)

What is range? Range: the set of output values (y-

coordinates)

Relation: a pair of quantities that are related in some way (a set of ordered pairs)

Page 3: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Definitions Continued What is a function? A function is a dependent

relationship between a first set (domain) and a second set (range), such that each member of the domain corresponds to exactly one member of the range. (i.e. NO x-values are repeated.)

Page 4: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Variable Reminders The independent/dependent variable is

the x-value The independent/dependent variable is

the y-value The independent variable is the

horizontal/vertical axis on an x-y plane The dependent variable is the

horizontal/vertical axis on an x-y plane

Page 5: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Determine whether the following correspondences are functions:Numbers:-3 9 3 2 4

Friday Night’s Date:

Juan Casandra

Boris Rebecca

Nelson HelgaBernie Natasha

YES!NO!

Page 6: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

You Do: Are these Correspondences Functions?Numbers:-6 36-2 4 2

Numbers:

-3 2 1 4 5 6 9 8

YES!NO!

Page 7: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Determine whether the relation is a function. If yes, identify the domain and range

{(2,10), (3,15), (4,20)}

Yes Domain: {2, 3, 4}. Range: {10, 15, 20}

{(-7,3), (-2,1), (-2,4), (0,7)}

No (the x-value of -2 repeats)

Page 8: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Determine whether the relation is a function. If yes, identify the domain and range

Domain

Range

-10 0

-8 2

-6 4

-4 6

-6 8No; -6 repeats

Domain

Range

-10 0

-8 2

-6 4

-4 6

-2 8Yes; D:{-10, -8, -6, -4, -2};

R:{0, 2, 4, 6, 8}

Page 9: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Testing for Functions Algebraically

Which of these is a function? A. x2 + y = 1 B. -x + y2 = 1

Do you know why?

Page 10: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Testing for Functions Algebraically

Which of these is a function? A. x2 + y = 1

Solve for y: y = -x2 + 1

No matter what I substitute for x, I will only get one y-value

Page 11: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Testing for Functions Algebraically

Which of these is a function? B. -x + y2 = 1

Solve for y:

If x = 3 for example, y = 2 or -2. So each x pairs with 2-different y’s. The x’s repeat—not a function.

y 1 x

Page 12: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Function Notation f(x) = y So f(x) = 3x + 2 means the same

thing as y = 3x + 2 f is just the name of the function

Page 13: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Evaluating a Function Let g(x) = -x2 + 4x + 1

A. Find g(2) B. Find g(t) C. Find g(x+2)

A. g(2) = 5 B. g(t) = -t2 + 4t + 1 C. g(x+2) = -x2 + 5

Page 14: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Interval Notation: Bounded Intervals

Notation Interval Type InequalityGraph

[a,b] Closed a x b [ ] a b

(a,b) Open a < x < b ( ) a b [a,b) Half-open a x < b [ ) Closed-left; a b

Open right (a,b] Half-open a < x b ( ]

Open-left a bClosed-right

Page 15: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Interval Notation: Unbounded Intervals

Notation Interval Type InequalityGraph

(-,b] Unbounded left x b ] Closed b

(-,b) Unbounded left x < b ) Open b

[a,) Unbounded right a x [ Closed a

(a,) Unbounded right a < x ( Open a

Page 16: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Domain: Graphical

[2,∞) (-∞,∞)

Page 17: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Domain: Graphical

(-∞,∞) [-3,∞)

Page 18: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Graphs: Are These Functions?

How Can You Tell?

Yes Yes

No No

The Vertical Line Test

Page 19: FUNCTIONS & GRAPHS 2.1 JMerrill, 2006 Revised 2008.

Are They Functions?Yes No

No Yes