functions - fcampena...prepared by: ms sonia tan domain = the set of all feasible values of x for...
TRANSCRIPT
FUNCTIONS
Prepared by: Ms Sonia Tan
1 a
b2
3
X Y
c
A function is a pairing that assigns to
each element of the set X exactly one
element of the set Y.
FUNCTION
Using Diagrams
Function Not a Function(only a relation)
1a
b
4
X Y
c
2
3
One to One Many to One One to Many
1 a
b
4
X Y
c
2
3d
Prepared by: Ms Sonia Tan
Using Ordered pairs
A relation is any set of ordered pairs.
A function is a set of ordered pairs such that no two
distinct ordered pairs have the same first coordinate.
{(1, a), (2, b), (3, d), (4, c)} {(1, a), (1, b), (2, b), (3, c)}
REAL LIFE EXAMPLES
State whether a function or not.
1. A student in a math class paired with his ID number.
2. A person paired with the type of car he or she has.
{(Annette, 123), (Judy, 049), (Sonia, 254)}
{ (Howard, Toyota), (Howard, BMW), (Janet, Honda)}
{(1, a), (2, a), (3, b), (4, b)}
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2y 2x x 5
Using Tables
Time 2006 2007 2008 2009
Population 5000 6800 9500 11400
An equation of the form y = f(x)
where each value of x , we get exactly one value of y.
Using Equations
Examples:
y = 2x + 4
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2y x
1 2
x
y
x
y
Using Graphs
x -2 -1 0 1
y
x 0 1 4
y 0
A Function
y = 2x + 4
Not a Function
•
•
•
•••
•
•
•
0 2 4 6
Graph of a function f is the set of all points (x,y) on the plane that
satisfy the equation y= f(x) .
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Vertical Line Test for Functions
If any vertical line intersects the graph of an
equation at exactly one point, then the equation
defines y as a function of x.
x
y
x
y
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Prepared by: Ms Sonia Tan
21. y x
22. y x 1
Exercises: State whether a function or not
using graphs
1xy.3 xy.43
21. y x 2
2. y x 1
function
x -2 -1 0 1 2
y 4 1 0 1 4
x -1 0 1 2 3
y 0 1 2 3 2
not a function
relationPrepared by: Ms Sonia Tan
function
x -1 0 1 2 2
y 0 -1 -2
x -8 -1 0 1 8
y -2 -1 0 1 2
1xy.3
2 3
31
xyxy.43
function
Prepared by: Ms Sonia Tan
FUNCTION VALUES
Given functions f(x) , f (a) means the value of
the function f at x = a. It is read as “f of a”
3x2x)x(f.Ex2
)2(f)h(f )2x(f
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Evaluate following:
Sum: f+g (x) = f(x) + g(x)
OPERATIONS ON FUNCTIONS
Given functions f(x) and g(x), we can have the
Difference: f-g (x) = f(x) - g(x)
Product: )x(g)x(f)x(gf
Quotient: )x(g
)x(f)x(
g
f
Composition of functions:
))x(g(f)x(gf ))x(f(g)x(fg
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EXAMPLE: Given f(x) = 2x – 5
g(x) = x2 + x – 6
Domain = the set
of all first
elements of the
ordered pairs
Example: {( -1, 1), ( 0, 0), ( 2, 4), ( 4, 16)}
Range = the set
of all second
elements of the
ordered pairs
DOMAIN AND RANGE
Give the domain and range of
{(1, 5), (3, 8), (5, 11), (7, 8), (9, -4)}
D = { -1, 0, 2, 4 } R = { 1, 0, 4, 16 }
Prepared by: Ms Sonia Tan
Domain = the set
of all feasible
values of x for
which y is defined
Range = the set of
all possible resulting
values of y or images
of x under the f
For mathematical equation y = f(x)
y = f(x) = – 3x +4
Df : R
Rf : R
y
•
x•
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y = h(x) =
Df : all x R such that
x – 2 0 or x 2
Rf : y 0
2x
Prepared by: Ms Sonia Tan
y = h(x) =
Df : all x R such that
Rf : y 0
4x2
2xor2x2x
4x4x04x222
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y = h(x) =
Df : all x R such that
Rf : y 0
2x4
2x22x4x
4x4x0x4
2
222
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Examples: f(x) = 2x + 5 a = 2 , b = 5
f(x) = -4x a = -4 , b = 0
f(x) = 10 a = 0 , b = 10
LINEAR FUNCTIONS
f(x) = ax + b where a and b are real numbers
The graph of a linear function is a straight line.
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Df : R
Rf : R
QUADRATIC FUNCTIONS
f(x) = ax2 + bx + c where a, b and c are real numbers
and .
The graph of quadratic function :
0a
V(h,k)
If a > 0, parabola upward. If a < 0, parabola downward.
V(h,k)
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f(x) = ax2 + bx + c
V = (h, k) is called the vertex of the parabola.
V(h,k)
V(h,k)
f(x) = a(x - h)2 + k
a2
bh
where and k = f(h)
x = h
x = hLowest point
Highest point
R:Dfky:R f ky:R f R:Df
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k = f(-1) = (-1)2 + 2(-1) -8 = -9
V = (-1, -9) f(x) = (x + 1)2 - 9
12
2
a2
bh
Examples: f(x) = x2 + 2x - 8 a = 1 , b = 2 , c = -8
V = (-1, -9)•
R:Df 9y:R f Prepared by: Ms Sonia Tan
Examples: f(x) = -x2 + x +2 a = -1 , b = 1 , c = 2
•
),(V4
9
21
4
92
21 )x()x(f
),(V4
9
21
4
9f y:R R:Df
Prepared by: Ms Sonia Tan
A Polynomial function in variable x of degree n
is of the form
k)x(f
01
2
2
1n
1n
n
n axaxaxaxa)x(P
and0a,Rs'awhere ni
.egerintnegativenonaisn
Constant function - polynomial of degree 0.
bax)x(f Linear function - polynomial of degree 1.
cbxax)x(f2 Quadratic function - polynomial of degree 2.
xx2x2)x(f2334 polynomial of degree 4
5xx3x2)x(g36 x5xx3x2)x(h
235
g and h are NOT POLYNOMIAL FUNCTIONS.Prepared by: Ms Sonia Tan
PIECE-WISE DEFINED FUNCTIONS
For y = -2x + 2
R:Df
0y:Rf
1xif1x
1xif2x2)x(f
2
When x = 1 , y = 0 (1, 0)
When x = 0 , y = 2 (0, 2)
For y = x2 + 1
When x = 1 , y = 2 (1, 2)
When x = 2 , y = 5 (2, 5)
•
y = -2x + 2
y = x2 + 1
•
•
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y = 2 for x < -1
R:Dg
y = - 3 for -1 x < 2
For y = x
When x = 2 , y = 2 (2, 2)
When x = 4 , y = 4 (4, 4)
•y = 2
y = -3•
•
2xifx
2x1if3
1xif2
)x(g
y = x
2yor3y:Rg
),2[3 Prepared by: Ms Sonia Tan
Prepared by: Ms Sonia Tan
Prepared by: Ms Sonia Tan
Prepared by: Ms Sonia Tan
Prepared by: Ms Sonia Tan
Prepared by: Ms Sonia Tan
Prepared by: Ms Sonia Tan