This document discusses functions and their representations. It begins by defining relations and functions, and providing examples of each. It then discusses different types of functions including linear, quadratic, constant, identity, absolute value, and piecewise functions. Examples are provided for each type. The document ends with exercises asking the reader to determine if given relations are functions, and to identify what type of function is being described.

1. General
Mathematics
G

2. General
Mathematics
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3. G General Mathematics II
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Representations of
Functions
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4. Objectives
define
functions and
related terms;
Statistics and
Probability
determine if
the given
relation
represents a
function;
Calculus
define
piece-wise
function;
and
Algebra
O J
B
represents
life situat
using func
including p
wise func
Calculus
E
>

5. define
functions and
related terms;
Statistics and
Probability
determine if
the given
relation
represents a
function;
Calculus
define
piece-wise
function;
and
Algebra
O
J
B

6. define
piece-wise
function;
and
Algebra
J
represents real-
life situations
using functions,
including piece-
wise functions.
Calculus
E
>
define
functions and
related terms;
Statistics and
Probability
O
determine if
the given
relation
represents a
function;
Calculus
B

7. define
piece-wise
function;
and
Algebra
J
d
;
determine if
the given
relation
represents a
function;
Calculus
B
represents real-
life situations
using functions,
including piece-
wise functions.
Calculus
E
define piece-
wise function;
and
Calculus
j
>

8. represents real-
life situations
using functions,
including piece-
wise functions.
E
Calculus
define piece-
wise function;
and
Calculus
j
>
define
piece-wise
function;
and
Algebra
J

9. Objectives
define
functions
and related
terms;
Statistics and
Probability
determine if
the given
relation
represents a
function;
Calculus
define
piece-wise
function;
and
Algebra
O J
B
represents
life situat
using func
including p
wise func
Calculus
E
>

10. R Relation
Sponsored
A relation is a set of ordered pairs. The
domain of a relation is the set of first
coordinates. The range is the set of
second coordinates.
Example of Relations T

11. coordinates. The range is the set of
second coordinates.
Example of Relations
1.{(1, 4), (2, 5), (3, 6), (4, 8)}
2.{(4, 2), (4, -2), (9, 3), (9,3)}
3.{(1, a), (1, b), (1, c), (1,d)}
T

12. F Function
Sponsored
A function is a relation in which each
element of the domain corresponds to
exactly one element of the range.
Examples of Functions
T

13. element of the domain corresponds to
exactly one element of the range.
Examples of Functions
1.{(1, 4), (2, 5), (3, 6), (4, 8)}
2.{(2, 1), (3, 1), (4, 1), (5,1)}
T

14. element of the domain corresponds to
exactly one element of the range.
Examples of Functions
1.{(1, 4), (2, 5), (3, 6), (4, 8)}
2.{(2, 1), (3, 1), (4, 1), (5,1)}
T
You have a message

15. A function is a relation in which each
element of the domain corresponds to
exactly one element of the range.
Examples of Functions
1.{(1, 4), (2, 5), (3, 6), (4, 8)}
2.{(2, 1), (3, 1), (4, 1), (5,1)}
T Relation & Function
Determine if the following relations
represent a function.
1. {(q, 0), (w, 1), (e, 2), (t, 3)}
2. {(-1, -2), (0, -2), (1, -2), (2, -2)}
3. {(1, 0), (1,1), (1, 2), (1, -2)}
4. {(x, 3), (y, 4), (z, 3), (w, 4)}

16. Sponsored
F Types of Functions

17. Sponsored
F Types of Functions
Linear Function
A function f is a linear
function if f(x) = mx + b,
where m and b are real
numbers, and m and f(x) are
not both equal to zero.

18. Sponsored
F Types of Functions

19. Sponsored
F Types of Functions
A quadratic function is any
equation of the form
f(x) = ax2+ bx + c where a, b,
and c are real numbers and
a ≠ 0.
Quadratic Function

20. Sponsored
F Types of Functions

21. Sponsored
F Types of Functions
Constant Function
A linear function f is a
constant function if
f(x) = mx + b, where m = 0
and b is any real number.
Thus, f(x) = b.

22. Sponsored
F Types of Functions

23. Sponsored
F Types of Functions
Identity Function
A linear function f is an
identity function if
f(x) = mx + b, where m = 1
and b = 0. Thus, f(x) = x.

24. Sponsored
F Types of Functions

25. Sponsored
F Types of Functions
Absolute Value Function
The function f is an absolute
value function if for all real
numbers x,
f(x) = x, for x ≥ 0
–x, for x ≤ 0

26. Sponsored
F Types of Functions

27. Sponsored
F Types of Functions
Piecewise Function
A piecewise function or a compound
function is a function defined by
multiple sub-functions, where each
sub-function applies to a certain
interval of the main function's
domain.

28. Sponsored
F Types of Functions

29. To sell more T-shirts, the class needs to charge a lower price as
indicated in the following table:
The price for which you can sell x printed T-shirts is called the
price function p(x). p(x) represents each data point in the table.
Target No. of
Shirt Sales
Price per T-shirt
500 P540
900 P460
1 300 P380
1 700 P300
2 100 P220
2 500 P140
Example 2
Step
Find

30. ce as
the
table.
Example 2
Step 1:
Find the slope m of the line using the
slope formula m = y2 – y1 / x2 – x1
Step 2:
Write th
of m an
of a line
Thus, th

31. Example 2
g the
2 – x1
Step 2:
Write the linear equation with two variables by substituting the values
of m and (x1, y1) to the formula y – y1 = m(x – x1)—the point-slope form
of a linear equation.
y – y1 = m(x – x1)
y – 540 = −15(x − 500)
y – 540 = − 15 x + 100
y = − 15 x + 640
y = 640 – 0.2x
Thus, the price function is p(x) = 640 – 0.2x.

32. E Example 3
30 mins ago
Find the dimensions of the largest
rectangular garden that can be
enclosed by 60 m of fencing.
T

33. Let x and y denote the lengths
of the sides of the garden. Then
the area A = xy must be given
its maximum value.
Example 3 | Solution
>

34. Express A in terms of a single variable, either x or y.
The total perimeter is 60 meters.
2x + 2y = 60
x + y = 30
y = 30 – x
Hence,
A = xy
A = x(30 – x)
A= 30x – x2
>
<

35. >
<
Write this equation in the vertex form by completing the
square.
A = –(x2 – 30x + 225) + 225
A = –(x – 15)2 + 225
The maximum area is 225 square meters.
Since x = 15 (the width) and 30 – x = 15 (the length), the
dimension that gives the maximum area is 15 meters by 15
meters.

36. E Example 3
30 mins ago
Find the dimensions of the largest
rectangular garden that can be
enclosed by 60 m of fencing.
T

37. E Example 4
30 mins ago
Sketch the graph of the given
piecewise function.
What is f(– 4)? What is f(2)?
T

38. Sketch the graph of the given
piecewise function. What is f(–
4)? What is f(2)?
f(x) = x + 2, if x ≥ 0
–x2+ 2, if x < 0
Example 4
>
{

39. To the right of the y-axis, the graph is a line
that has a slope of 1 and y-intercept of 2. To
the left of the y-axis, the graph of the
function is a parabola that opens downward
and whose vertex is (0, 2).
Example 4 | Solution
>
<

40. To sketch the graph of
the function, you can
lightly draw both graphs.
Then darken the portion
of the graph that
represents the function.
Example 4 | Solution
>
<

41. To find the value of the function when x = – 4, use
the second equation.
f(– 4) = – (– 4)2+ 2 = – 16 + 2 = – 14
Example 4 | Solution
>
<

42. To find the value of the function when x = 2, use
the first equation.
f(2) = 2 + 2 = 4
Example 4 | Solution
>
<

43. Exercises
Statistics and
Probability
Calculus Algebra
E C
X
4
Algebra
E
>

44. a Determine whether or not each
relation is a function. Give the
domain and range of each relation.
1. {(2, 3), (4, 5), (6, 6)}
2. {(5, 1), (5, 2), (5, 3)}
3. {(6, 7), (6, 8), (7, 7), (7, 8)}

45. Exercises
Statistics and
Probability
Calculus Algebra
E C
X
4
Algebra
E
>

46. Tell whether the function described in each of
the following is a linear function, a constant
function, an identity function, an absolute
value function, or a piecewise function.
4. 5.
b

47. Exercises
Statistics and
Probability
Calculus Algebra
E C
X
4
Algebra
E
>

48. Tell whether the function described in each of
the following is a linear function, a constant
function, an identity function, an absolute
value function, or a piecewise function.
4. 5.
c

49. Exercises
4
Statistics and
Probability
Calculus Algebra
E C
X
Algebra
E
>

50. Exercises
Statistics and
Probability
Calculus Algebra
E C
X
4
Algebra
E
>
G General Mathematics II
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