LESSONS 1-3 Gen Math. functions and relationsppt

gen math

2. Relations
and
Functions

3. Relation-Any set of ordered
pairs.
domain-first coordinates,
input, independent variable
range-second coordinates,
output, dependent variable
Definitions

4. Example
•People and their heights
•Grades in General Mathematics

5. Example
Names
Peter
Krisha
Fay
Abby
Earl
Clara
Job
Karl
Audrey
Grades
95
94
93
92
95
97
94
90
98

6. Function-a type of relation
where there is exactly one
outputfor every input. For
every xthere is exactly one y.

7. xy
Job
Karl
96
97
xy
Job
Karl
96
97
Not a FunctionFunction

8. Vertical Line Test -Functions
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
Function

9. Vertical Line Test -Functions
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
FunctionFunction

10. Vertical Line Test -Functions
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
FunctionFunction Not a
Function

11. Vertical Line Test -Functions
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
FunctionFunction Not a
Function
Function

12. Vertical Line Test -Functions
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
FunctionFunction Not a
Function
Function
Not a
Function
Function
Not a
Function
Not a
Function

13. Tell whether the relation below is a function.
1)
2)
3)
4)
inputoutput
0
1
5
2
3
yx
-3
-3
-3
-3
-1
0
1
2
x
y
Function
Not a
Function
Not a
Function
inputoutput
-2
-1
0
3
4
5
6
Not a
Function

14. Conclusion and Definition
•Not every relation is a function.
•Every function is a relation.

15. 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.

16. yx
-2
-1
0
1
2
-4
-2
0
2
4
y = 2x
inputoutput
-2
-1
-4
-2
0
1
0
2
2 4
x-y chartmapping

17. QUADRATIC FUNCTION
A quadratic function is any
equation of the form f(x) =
ax
2
+ bx+ c where a, b, and c
are real numbers and a ≠0.

18. y = x
2
+ 1
yx
-2
-1
0
1
2
5
2
1
2
5
inputoutput
-2
-1 5
20
1
1
2
x-y chartmapping

19. 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.
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.

20. 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.

22. yx
-2
-1
0
1
2
-4
-2
0
2
4
y = 2x
inputoutput
-2
-1
-4
-2
0
1
0
2
2 4
x-y chartmapping

23. Many-to-one Function
if there areyvalues that have more
than onexvalue mapped onto them.
One-to-many (not a function)

24. xy
Job
Karl
96
97
xy
Job
Karl
96
97
Not a FunctionFunction
One-to-many Many-to-one

25. Lesson 2 EVALUATION OF FUNCTIONS

26. EVALUATING FUNCTIONS
LAW OF SUBSTITUTION
If a + x = b and x = c, then a+ c= b
Example 1
If f(x) = x + 8, evaluate each.
a.f(4)
b.f(–2)
c.f(–x)
d.f(x + 3)

27. Function Notationy 2x 3 f (x) 2x 3 when x 1, y 5 when x 2, y 7 when x 3, y 9 when x 4, y 11 f (1) 5 f(2) 7 f(3) 9 f(4) 11 f( 4) 5

28. 2
g(x) x h(x) 3x 2 Evaluate the following.1) g(4) 2) h( 2) 3) g( 3) 4) h(5) 5) h(4) g(1) 6) h( 5) g( 2)    7) g h(3) 8) h g(2) 16 8 9 13 10  1 1 17  4 68 g ( 7 ) 4 9 h ( 4 ) 1 0

29. Lesson 3 OPERATIONS ON FUNCTIONS

30. Lesson Objectives
At the end of the lesson, the students must
be able to:
•find the sum of functions;
•determine the difference between
functions;
•identify the product of functions;
•find the quotient between functions; and
•determine the composite of a function.

31. ADDITION OF FUNCTIONS
Let f and g be any two functions.
The sum f + g is a function whose
domains are the set of all real numbers
common to the domain of f and g, and
defined as follows:
(f + g)(x) = f(x) + g(x)

32. SUBTRACTION OF FUNCTIONS
Let f and g be any two functions.
The differencef –g is a function whose
domains are the set of all real numbers
common to the domain of f and g, and
defined as follows:
(f –g)(x) = f(x) –g(x)

33. MULTIPLICATION OF FUNCTIONS
Let f and g be any two functions.
The product fgis a function whose
domains are the set of all real numbers
common to the domain of f and g, and
defined as follows:
(fg)(x) = f(x) · g(x)

34. DIVISION OF FUNCTIONS
Let f and g be any two functions.
The quotientf/g is a function whose domains
are the set of all real numbers common to the
domain of f and g, and defined as follows: ,
where g(x) ≠ 0.f
g
x()=
fx()
gx()

35. Example 1
If f (x) = 3x –2 and
g (x) = x
2
+ 2x –3,
1. find (f + g) (x)
2. Find (f –g)(x)
3. Solve (fg)(x)
4. Find the quotient of f(x) and g(x)