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Functions. August 18 2013. Definition of a function. A function takes an element from a set and maps it to a UNIQUE element in another set. Function terminology. f maps R to Z. f. R. Z. Co-domain. Domain. f(4.3). 4. 4.3. Pre-image of 4. Image of 4.3. “a” “bb“ “cccc” “dd” “e”. - PowerPoint PPT Presentation
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FunctionsAugust 18 2013
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Definition of a function
A function takes an element from a set and maps it to a UNIQUE element in another set
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Function terminology
R Zf
4.3 4
Domain Co-domain
Pre-image of 4 Image of 4.3
f maps R to Z
f(4.3)
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More functions
1
2
3
4
5
“a”
“bb“
“cccc”
“dd”
“e”
A string length function
A
B
C
D
F
Alice
Bob
Chris
Dave
Emma
A class grade function
Domain Co-domainA pre-image
of 1
The imageof A
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Even more functions
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2
3
4
5
“a”
“bb“
“cccc”
“dd”
“e”
Not a valid function!Also not a valid function!
1
2
3
4
5
a
e
i
o
u
Some function…
Range
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Function arithmetic
Let f1(x) = 2x
Let f2(x) = x2
f1+f2 = (f1+f2)(x) = f1(x)+f2(x) = 2x+x2
f1*f2 = (f1*f2)(x) = f1(x)*f2(x) = 2x*x2 = 2x3
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One-to-one functions
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2
3
4
5
a
e
i
o
A one-to-one function
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2
3
4
5
a
e
i
o
A function that is not one-to-one
A function is one-to-one if each element in the co-domain has a unique pre-image
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More on one-to-one
Injective is synonymous with one-to-one “A function is injective”
A function is an injection if it is one-to-one
Note that there can be un-used elements in the co-domain
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2
3
4
5
a
e
i
o
A one-to-one function
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Onto functions
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2
3
4
5
a
e
i
o
A function that is not onto
A function is onto if each element in the co-domain is an image of some pre-image
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2
3
4
a
e
i
o
u
An onto function
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1
2
3
4
a
e
i
o
u
An onto function
More on onto
Surjective is synonymous with onto “A function is surjective”
A function is an surjection if it is onto
Note that there can be multiply used elements in the co-domain
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Onto vs. one-to-oneAre the following functions onto, one-to-one, both, or neither?
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2
3
4
a
b
c
1
2
3
a
b
c
d
1
2
3
4
a
b
c
d
1
2
3
4
a
b
c
d
1
2
3
4
a
b
c
1-to-1, not onto
Onto, not 1-to-1
Both 1-to-1 and onto Not a valid function
Neither 1-to-1 nor onto
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Bijections
Consider a function that isboth one-to-one and onto:
Such a function is a one-to-one correspondence, or a bijection
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2
3
4
a
b
c
d
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Identity functions
A function such that the image and the pre-image are ALWAYS equal
f(x) = 1*x
f(x) = x + 0
The domain and the co-domain must be the same set
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Inverse functions
R Rf
4.3 8.6
Let f(x) = 2*x
f-1
f(4.3)
f-1(8.6)
Then f-1(x) = x/2
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More on inverse functions
Can we define the inverse of the following functions?
An inverse function can ONLY be done defined on a bijection
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2
3
4
a
b
c
1
2
3
a
b
c
d
What is f-1(2)?Not onto!
What is f-1(2)?Not 1-to-1!
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Compositions of functions
Let (f ○ g)(x) = f(g(x))
Let f(x) = 2x+3 Let g(x) = 3x+2
g(1) = 5, f(5) = 13
Thus, (f ○ g)(1) = f(g(1)) = 13
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Compositions of functions
g f
f ○ g
g(a) f(a)
(f ○ g)(a)
g(a)f(g(a))a
A B C
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Compositions of functions
g f
f ○ g
g(1) f(5)
(f ○ g)(1)
g(1)=5
f(g(1))=131
R R R
Let f(x) = 2x+3 Let g(x) = 3x+2
f(g(x)) = 2(3x+2)+3 = 6x+7
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Compositions of functions
Does f(g(x)) = g(f(x))?
Let f(x) = 2x+3 Let g(x) = 3x+2
f(g(x)) = 2(3x+2)+3 = 6x+7
g(f(x)) = 3(2x+3)+2 = 6x+11
Function composition is not commutative!
Not equal!
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Graphs of functions
Let f(x)=2x+1
Plot (x, f(x))
This is a plotof f(x)
x=1
f(x)=3
x=2
f(x)=5
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Useful functions
Floor: x means take the greatest integer less than or equal to the number
Ceiling: x means take the lowest integer greater than or equal to the number
round(x) = floor(x+0.5)
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Rosen, question 8 (§1.8)
Find these values
1.11.1-0.1-0.12.99-2.99½+½½ + ½ + ½
12-103-2½+1 = 3/2 = 10 + 1 + ½ = 3/2 = 2
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Ceiling and floor properties
Let n be an integer(1a) x = n if and only if n ≤ x < n+1(1b) x = n if and only if n-1 < x ≤ n(1c) x = n if and only if x-1 < n ≤ x(1d) x = n if and only if x ≤ n < x+1(2) x-1 < x ≤ x ≤ = x < x+1 (3a) -x = - x(3b) -x = - x(4a) x+n = x+n (4b) x+n = x+n
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Ceiling property proof
Prove rule 4a: x+n = x+n Where n is an integer Will use rule 1a: x = n if and only if n ≤ x <
n+1
Direct proof! Let m = x Thus, m ≤ x < m+1 (by rule 1a) Add n to both sides: m+n ≤ x+n < m+n+1 By rule 4a, m+n = x+n Since m = x, m+n also equals x+n Thus, x+n = m+n = x+n
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Factorial
Factorial is denoted by n!
n! = n * (n-1) * (n-2) * … * 2 * 1
Thus, 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720
Note that 0! is defined to equal 1
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Proving function problems
Rosen, question 32, §1.8
Let f be a function from A to B, and let S and T be subsets of A. Show that
)()()( )
)()()( )
TfSfTSfb
TfSfTSfa
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Proving function problems
Rosen, question 32 (a): f(SUT) = f(S) U f(T)Will show that each side is a subset of the otherTwo cases!Show that f(SUT) f(S) U f(T)
Let b f(SUT). Thus, b=f(a) for some aS U T Either aS, in which case bf(S) Or aT, in which case bf(T) Thus, bf(S) U f(T)
Show that f(S) U f(T) f(S U T) Let b f(S) U f(T) Either b f(S) or b f(T) (or both!) Thus, b = f(a) for some a S or some a T In either case, b = f(a) for some a S U T
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Proving function problems
Rosen, question 32 (b): f(S∩T) f(S) ∩ f(T)
Let b f(S∩T). Then b = f(a) for some a S∩T
This implies that a S and a TThus, b f(S) and b f(T)
Therefore, b f(S) ∩ f(T)
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Proving function problems
Rosen, question 62, §1.8
Let f be an invertible function from Y to Z
Let g be an invertible function from X to Y
Show that the inverse of f○g is: (f○g)-1 = g-1 ○ f-1
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Proving function problems
Rosen, question 62, §1.8Thus, we want to show, for all zZ and xX
The second equality is similar
xxgffg
zzfggf
)(
)(11
11
z
zff
zfggf
zfggf
zfggfzfggf
)(
)(
)(
)()(
1
11
11
1111
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Quick survey
I felt I understood the material in this slide set…
a) Very wellb) With some review, I’ll be goodc) Not reallyd) Not at all
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Quick survey
The pace of the lecture for this slide set was…
a) Fastb) About rightc) A little slowd) Too slow
34
Quick survey
How interesting was the material in this slide set? Be honest!
a) Wow! That was SOOOOOO cool!b) Somewhat interestingc) Rather bortingd) Zzzzzzzzzzz