2.2, 2.3 Functions

• Function is a corresponding between 2 sets: domain and range, such that each member of the domain correspond to exactly one member of the range

• For example:– Each person corresponds to his or her biological mother– Each person corresponds to his or her weight– Each natural number (1, 2, 3, 4…) corresponds to the

square of that number (1, 4, 9, 16…)

Domain Rangecorrespondence

• In a set of ordered pairs, the domain is the set of all first coordinate (x), and the range is the set of all second coordinate (y)

• For example: {(1,-1), (2,-2),(3,-3),(4,-4)}– The domain is {1,2,3,4} – The range is {-1,-2,-3,-4}

• Determine whether each of the following is a function. If yes, list the domain and the range

1) {(Sue, 18 years old), (Peter,19 years old),(Kim, 16 years old), (Sue, 20 years old)}– This is not a function because Sue corresponds to

two numbers: 18 and 20 years olds

2) {(1,3), (2,3), (3,4)}– This is a function. The domain is {1,2,3}, and the

range is {3,4}

3) y = x3

– This is a function. The domain is {1,2,3,4,…} and the range is {1,8,27,64…}

NO, because z corresponds to both X and Z

YES, each element in the domain corresponds to only one element in the range

• Determine a function by The Vertical-Line Test: if the vertical line cross the graph more than once, then the graph is not a function

yesno

yes

• Function notation: f(x) read f of xEx: f(x) = 2xImagine this function is a change machine. If we put $1

bill in the machine, it will give out 2 coins of 50cents. x (number of dollar bills) f(x) number of coins

INPUT OUTPUT 2 4 3 6

4 8 Find f(6), f(a + 1) if f(x) = 2x F(6) = 2 * 6 =12 F(a + 1) = 2 * (a + 1) = 2a + 2

Ex2: F(n) = 3n2 – 2nFind f(0), f(-1), f(2a), 3f(a)• F(0) = 3*02 – 2*0 = 0 - 0 = 0• F(-1)= 3(-1 )2 – 2(-1) = 3 + 2 = 5• F(2a) = 3(2a)2 – 2(2a) = 3 * 4a2 - 4a = 12a2 – 4a• 3 f(a) = ?F(a) = 3*a2 – 2*a = 3a2 – 2a 3f(a) = 3 (3a2 – 2a ) = 9a2 – 6a

Find the domain and the range for each function

Domain: all real numbers (-∞, ∞)

Range: all real numbers (-∞, ∞)

Domain: all real numbers (-∞, ∞)

Range: [-4, ∞)

Domain: (-5, 4)

Range: (-5, 5] Range: [-2, 2)

Domain [-3, 3)

More problems with domain1) f(x) = x + 1

Domain is (-∞,∞) interval notation

2) f(x) = 2x2 + 1

x + 2

Domain is all real numbers except -2

(-∞,-2) U (-2, ∞) interval notation

Credit card debt in the US from 1992 through

1999 is modeled by this equation:

y = 47.3x + 281 (in billion)

where x = 0 represents for 1992

a) Approximate the credit card debt in 1992, 1993, and 1999 using the equation

b) Graph the linear equation using the information from a

c) Use the graph to approximate the credit card debt in 1996

y = 47.3x + 281 (in billion dollars)

For 1992, x = 0So y = 47.3(0) + 281 = 281 For 1993, x = 1So y = 47.3(1) + 281 = 328.3

For 1999, x = 7So y = 47.3(7) + 281 = 612.1

For 1996, look at the graph, we have y = 470 billion dollars

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