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The function as a machine…
We will try to represent mathematical
relations as machines with an input
and an output, and that the output is
related to the input by some rule.
Determine if this machine produces
a function…
Determine if this machine produces
a function…
Determine if this machine produces
a function…
Determine if this machine produces
a function…
Is it a function or not?
1.A jeepney and its plate number
2.A student and his ID number
3.A pen and the color of its ink
4. A P50 bill and its serial number
5. A Mathematics teacher and his
students
A relation is a set of
ordered pairs. The
domain of the
relation is the set of
first coordinates. The
range is the set of
second coordinates.
A function is a
relation in which
each element of the
domain corresponds
to exactly one
element of the range.
FUNCTIONS AND RELATIONS AS A
SET OF ORDERED PAIRS
Which of the following
relations are functions?
(a)f = {(1,2), (2,2), (3,5), (4,5)}
(b)g = {(1,3), (1,4), (2,5), (2,6),
(3,7)}
(c)h = {(1,3), (2,6), (3,9), …,
(n, 3n),…}
FUNCTION
RELATION
FUNCTION
FUNCTIONS AND RELATIONS
AS A TABLE OF VALUES
RELATION
FUNCTION
FUNCTION
x 1 2 3 4
f(x) 5 7 9 11
x -1 0 -1 2 3
f(x) 8 3 -6 4 11
x 0 0.5 1.5 2.5 3.5 4.5 5
f(x) 0 1.4 4.2 7 4.2 1.4 0
FUNCTIONS AND RELATIONS
AS A MAPPING DIAGRAM
Which of the following diagram represents a function?
FUNCTION FUNCTION
RELATION
ONE-TO-ONE
FUNCTION
MANY-TO-
ONE
FUNCTION
ONE-TO-MANY
FUNCTIONS AND RELATIONS AS A
GRAPH IN THE CARTESIAN PLANE
Which of the following graphs represent a
function?
FUNCTION FUNCTION
FUNCTION
RELATION
RELATION
THE VERTICAL LINE TEST
A graph represents a function
if and only if each vertical line
intersects the graph at most once.
FUNCTION? RELATION?
FUNCTION
FUNCTION
FUNCTION
FUNCTION
RELATION
All functions are
relation, but not all
relations are
function.

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Functions as models

  • 1.
  • 2. The function as a machine… We will try to represent mathematical relations as machines with an input and an output, and that the output is related to the input by some rule.
  • 3. Determine if this machine produces a function…
  • 4. Determine if this machine produces a function…
  • 5. Determine if this machine produces a function…
  • 6. Determine if this machine produces a function…
  • 7. Is it a function or not? 1.A jeepney and its plate number 2.A student and his ID number 3.A pen and the color of its ink 4. A P50 bill and its serial number 5. A Mathematics teacher and his students
  • 8. A relation is a set of ordered pairs. The domain of the relation is the set of first coordinates. The range is the set of second coordinates. A function is a relation in which each element of the domain corresponds to exactly one element of the range.
  • 9. FUNCTIONS AND RELATIONS AS A SET OF ORDERED PAIRS Which of the following relations are functions? (a)f = {(1,2), (2,2), (3,5), (4,5)} (b)g = {(1,3), (1,4), (2,5), (2,6), (3,7)} (c)h = {(1,3), (2,6), (3,9), …, (n, 3n),…} FUNCTION RELATION FUNCTION
  • 10. FUNCTIONS AND RELATIONS AS A TABLE OF VALUES RELATION FUNCTION FUNCTION x 1 2 3 4 f(x) 5 7 9 11 x -1 0 -1 2 3 f(x) 8 3 -6 4 11 x 0 0.5 1.5 2.5 3.5 4.5 5 f(x) 0 1.4 4.2 7 4.2 1.4 0
  • 11. FUNCTIONS AND RELATIONS AS A MAPPING DIAGRAM Which of the following diagram represents a function? FUNCTION FUNCTION RELATION ONE-TO-ONE FUNCTION MANY-TO- ONE FUNCTION ONE-TO-MANY
  • 12. FUNCTIONS AND RELATIONS AS A GRAPH IN THE CARTESIAN PLANE Which of the following graphs represent a function? FUNCTION FUNCTION FUNCTION RELATION RELATION
  • 13. THE VERTICAL LINE TEST A graph represents a function if and only if each vertical line intersects the graph at most once.
  • 15.
  • 16. All functions are relation, but not all relations are function.