Functions

1
Functions

• Reading Chapter 6 (94 107) from the text book

2
What is a function?

• A function is an input/output rule defined on
• some set.
• Example the rule that takes as input a
• student in this class, and produces as output
• that students age.
• Intuitively a function from A to B is a way to
• transform each element of A into an element
  of B

3
What is a function?

• Two conditions must be satisfied
• - Each valid input item must produce some
  
• output.
• -The output depends only on the input, and
  
• not on any other extra factors. That is,
• identical input will always produce the
  same
• output.

4
Relations and functions

• A function can be thought of as a special
  kind
• of relation we get this relation by simply
• taking the set of all pairs
• (input, output)
• for every possible input.
• Formally a function from A to B is a relation on
  
• A B such that for all x ? A there exists
  exactly
• one element y of B such that (x, y) is in the
  relation

5
Functions as lists of pairs

• Let A 1, 2, 3 and B a, b, c, d, e.
  Consider the relation
• g ? A B g (1, d), (2, c), (3,
  c)
• This is a function each element of A is paired
  with a single element of B (both 2 and 3 are
  paired with c but thats allowed).

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Functions as lists of pairs

• By contrast h (1, a), (3, c), (1, d)
• is not a function, because 1 is paired with two
  things, and 2 isnt paired with anything.
• Exercise Which of the following relations
  on a, b, c 1, 2, 3 are functions
• R1 (a, 1), (a, 2), (b, 3), (c,
  2)
• R2 (a, 1), (b, 2), (c, 1)
• R3 (a, 1), (c, 2)

7
Function notation

• When we have a function such as g above, we
  often write it in the following way
• g(1) d g(2) c g(3) d
• Read g of 1 is equal to d and so on. This
  notation is only used for functions and not for
  more general relations.

8
Arrow diagram

• Arrow diagrams are used to depict functions
• for which input sets are finite.
• Example Let A1, 2, 3 and B1, 2, 3, 4. Let
  f A B, f(1)3, f(2)2, f(3)2. Draw an
  arrow diagram for this function.
• A
  B

1
1
2
2
3
3
4
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Functions and formulas

• It is actually rare to specify a function as a
  list of pairs. More usually, we give a rule or
  formula that makes it possible to work out the
  value (output) of a function on any given element
  of its domain(input).
• For instance, we can define a function
• f ?RR by the formula f(x) x2 3

10
Functions and formulas

• To work out the value of the function we just
  substitute in the formula
• f(4) 42 3 16 3 19
• or in another way of writing the same thing
• (4, 19) ? f. We could also write
• f (x, x2 3) ? x ? R

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Important numerical functions

• Functions from number systems (such as N, Q, or
  R) to themselves, or to other number systems are
  very important, and some of them have special
  names.
• Well deal with functions from R to R for
  convenience, but the names are the same in other
  cases.

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Important numerical functions

• A linear function is one whose value is obtained
  by multiplying its input by a fixed amount, and
  then adding or subtracting a fixed amount from
  the result.
• For example
• f(x) 14x - 3
• In a quadratic function we may also add or
  subtract a fixed multiple of the square of the
  input.

13
Important numerical functions

• For example g(x) 3x2 - 7x 2
• More generally, a polynomial function adds
• together multiples of fixed powers of the
• input
• h(x) -2x5 5x3 - 3x2 17x 1
• In an exponential function we use the input as
• an exponent over a fixed base s(x) 2x

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Domain, codomain and range

• The domain of a function is the set of its
  allowed inputs.
• The co-domain of a function is the set in which
  we guarantee its outputs lie.
• The range of a function is the actual set of its
  outputs.

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Domain, codomain and range

• To see the difference between the last two
• concepts consider f R ? R defined by
• f(x) x2
• Its co-domain is R but its range is the set of
• non-negative real numbers.

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Terminology

• Let f be a function from A to B. Let x ? A
• The image of x under f, denoted by f(x), is the
  only element of B such that (x, f(x)) ? f
• Intuitively, f transforms x into f(x)
• The domain of f is A
• the codomain of f is B
• The range of f is f(x) x ? A

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Identity and constant functions

• If the output of a function is always the same,
• no matter what the input is, then the function
• is called a constant function.
• Example f R?R defined by f(x) 42 for all x.
• Also, given any set A, there is a special
  function
• called the identity function on A, and usually
• denoted iA, which has no effect on its input.
  That is
• iA(x) x for all x ? A.

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Equality of functions

• If f and g are functions from A to B then f g
• if, for every a ? A, f(a) g(a).
• For example f, g R ? R
• f(x) (x 1)2 - 2x
• g(x) x2 1
• These are equal functions because for any
• real number x
• f(x) (x 1)2 - 2x x2 2x 1 - 2x x2 1
  g(x)

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One to One functions

• A function, f, is called one to one or one-one
• if, whenever a ? b, then f(a) ? f(b). That is,
  for
• any two distinct inputs to the function, the
• outputs are also distinct.
• Put another way, if f(a) f(b), then necessarily
  
• ab.
• For example, the function g R?R defined by
• g(x) 3x 1 is one-one.

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One to One functions

• one to one because
• If g(a) g(b) then 3a 1 3b 1,
• and so 3a 3b,
• and finally a b.
• On the other hand, the function f R?R defined
  by f(x) x2x2 is not one to one because
• f(0) 2 f(-1) but 0 ? -1.

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Onto functions

• A function, g A ? B, is called onto or
• if, for every b ? B there is at least one a ? A
• such that g(a) b.
• In other words, the range of an onto function
• is equal to its co-domain.
• The function h R?R defined by h(x) 5x -2
• is onto because, for given b ?R we can take
• a (b 2)/5 and then
• h(a) 5a - 2 5(b 2)/5 - 2 b 2 - 2 b

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One-One correspondences

• A function that is both one to one and onto is
  called a one-one correspondence
• If f A ? B is a one-one correspondence then,
  for every b ? B there is exactly one a ? A such
• that f(a) b.
• The identity function on any set A is always
• one-one correspondence. Consider f J ?N
  defined as follows
• f(x)

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One-One correspondences

• Then, f is a one-one correspondence. To see this,
  let an arbitrary natural number n be given.
• If n 2k is even, then n f(k), while if n
  2k 1 is odd, then n f(-k). So, f is onto.
• On the other hand, f is also one to one, since
• from the value of f(a) we can uniquely determine
  a
• (if the value is even, a is f(a)/2 and if the
  value is odd, a is -(f(a) - 1)/2.

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One to One, Onto, One-One correspondence

• Let f be a function from A to B
• f is one to one iff for all x, y ? A, if x y,
  then
• f(x) f(y)
• f is onto iff for all y ? B, there exists x ? A
  such
• that f(x) y
• f is one-one correspondence iff f is both
• one to one and onto

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Exercise

• For each of the following functions from
• a, b, c, d to 1, 2, 3, 4, decide whether
  it is one to one and/or onto
• R1 (a, 1), (d, 3), (b, 2), (c, 4)
• R2 (a, 1), (b, 2), (c, 3), (d, 2)
• Solution
• R1 is both one to one and onto
• R2 is neither one to one nor onto

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Composition of functions

• If we think of functions as input-output rules
  then it is natural to string them together, using
  the output of one rule as input to the next.
• This operation is known as composition of
  functions and is defined formally as follows
  Suppose that f A ? B and g B ? C are
  functions (note that the domain of g its input,
  matches the co-domain of f its possible
  outputs.)

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Composition of functions

• Then, given a ? A we can take b f(a) and
• then c g(b). This association
• a ? b ? c i.e. a ? f(a) ? g(f(a))
• defines a function h A ? C called the
  composite or composition of f and g.
• We write h g ?f.
• Very important the function that is applied
  second is written first.

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Examples of composition

• Let f, g R?R be given by f(x) 3x - 1 and
• g(x) x2 1. Then (g ?f)(x) (3x - 1)2 1
• In other words, the composite of g and f is
• really obtained by substituting the value of f
• for the argument of g.
• Note also that since, in this case, the domain
  and co-domain are the same, we could form the
  other composite
• (f ?g)(x) f(g(x)) f(x2 1) 3(x2
  1) - 1
• And that this is not equal to g ?f.

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Rules of composition

• Let f A ? B, g B ? C and h C ? D be
  functions. Then
• f ?iA f iB ?f
• h ?(g ?f) (h ?g) ?f
• Also, if both f and g are one-one then so is
• g ? f. Likewise, if both are onto so is f ?
  g.

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Inverse of a function

• Remember that we can think of a function
• f A ? B as a relation, that is, as a subset
  of
• A B. Specifically
• f (a, f(a)) ? a ? A
• For a general relation, we have defined the
• notion of its inverse obtained by flipping all
  
• the ordered pairs belonging to it
• f-1 (f(a), a) ? a ? A

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Inverse of a function

• It seems natural to ask What conditions, if
• any, do we need to impose on f in order to
• ensure that f-1 is a function?.
• Note first that, if f-1 does happen to be a
  function, then it will have the following
  properties for all a ? A, f-1(f(a)) a, and for
  all b ? B, f(f-1(b)) b.
• Theorem Let f A ? B be a function. The inverse
  relation f-1 B ? A is a function if and only if
  f is a one-to-one correspondence.

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Inverse relations and invertible functions

• Let R be a relation on A B
• The inverse of R, denoted by R-1, is defined as
  follows
• R-1 (y, x) x, y ?
  R
• Let f be a function from A to B
• As f is a relation, we can form the inverse
  relation f-1
• If f-1 is also a function, we say that f is
  invertible and call f-1 the inverse function