Mathematics for Computing

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Mathematics for Computing
Lecture 8 Functions Dr Andrew
Purkiss-Trew Cancer Research UK e-mail
a.purkiss_at_mail.cryst.bbk.ac.uk
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Functions

• Functions
• What is a function?
• Range and other rules
• Composite functions
• Inverse of function

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Formula

• Functions relate two sets of numbers
• Each x gives a value of yso in the first
  function, x 2, gives y 18.In the second
  function, x 1, gives y -1.

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General form

• General formuse f,g or h to represent
  function.

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Functions and Sets

• Definition For sets X and Y, A function from X
  to Y is a rule that assigns each element of X to
  a single element of Y
• X is the domain, Y is the codomain
• If is any element of X. Then each element of
  Y assigned to is called the image of and
  written

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Example

x
x2
f
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Range

• If fX?Y is a function then the range isy?Y y
  f(x) for some or all x?X
• Example The range of f is yy?0

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Another example

• A 1,3,5, B2,4,6,8
• fA ?B, f(1)2, f(3)6, f(5)2
• The range of f 2,6

1 3 5 A
2 4 6 8 B
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More definitions

• OntoA function is onto if its range is equal to
  its codomain.
• One-to-oneA function is one-to-one if no two
  distinct elements of the domain have the same
  image.

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Examples of definitions
Not one-to-one Not onto
One-to-one Not onto
Not one-to-one Onto
One-to-one Onto
Not a function
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Composite functions

• Composite function link two functions together
• Let A,B and C be arbitrary setsfA?B and gB?C
• Input is xx?A and output g(f(x))?C

f(x)
g(f(x))
x
f
g
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Composite Functions 2

• Formal definitionLet fA?B and gB?C. The
  composite function of f and g is g o f A?C,
  (g o f)(x) g(f(x))

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Composite Function Example

• fR?R, f(x) x2 and gR?R, g(x) 2x 1
• f o g R?R, (f o g)(x) f(g(x))
  f(2x1) (2x1)2
• g o f R?R, (g o f)(x) g(f(x))
  g(x2) 2x21

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Identity and Inverse

• IdentityIA?A, i(x)x
• Inverse of a function is the function that
  reverses the effect of the function. It is
  represented by f 1 for the function f

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Inverse 2

• Let fA?B and gB?A be functionsIf g o f A?A
  is the identity function on A and if f o g B?B
  is the identity function on B, then f is the
  inverse of g (and g is the inverse of f )

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