Transcript Lesson 2.2, 2.3

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
correspondence
Domain
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…)
• 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
yes
no
yes
• Function notation: f(x) read f of x
Ex: f(x) = 2x
Imagine 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 – 2n
Find 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 (-∞, ∞)
Domain: all real
numbers (-∞, ∞)
Range: all real
numbers (-∞, ∞)
Range: [-4, ∞)
Domain: (-5, 4)
Domain [-3, 3)
Range: (-5, 5]
Range: [-2, 2)
More problems with domain
1) 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)
b)
c)
Approximate the credit card debt in 1992, 1993, and
1999 using the equation
Graph the linear equation using the information from a
Use the graph to approximate the credit card debt in
1996
y = 47.3x + 281 (in billion dollars)
For 1992, x = 0
So y = 47.3(0) + 281 = 281
700
600
500
For 1993, x = 1
So y = 47.3(1) + 281 = 328.3
400
300
200
For 1999, x = 7
So y = 47.3(7) + 281 = 612.1
100
0
0
For 1996, look at the graph, we
have y = 470 billion dollars
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