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FUNCTIONS & GRAPHS2.1
JMerrill, 2006Revised 2008
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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)
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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.)
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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
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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!
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You Do: Are these Correspondences Functions?Numbers:-6 36-2 4 2
Numbers:
-3 2 1 4 5 6 9 8
YES!NO!
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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)
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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}
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Testing for Functions Algebraically
Which of these is a function? A. x2 + y = 1 B. -x + y2 = 1
Do you know why?
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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
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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
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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
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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
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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
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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
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Domain: Graphical
[2,∞) (-∞,∞)
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Domain: Graphical
(-∞,∞) [-3,∞)
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Graphs: Are These Functions?
How Can You Tell?
Yes Yes
No No
The Vertical Line Test
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Are They Functions?Yes No
No Yes