functions relation – a set of ( x, y ) points function – a set of ( x, y ) points where there is...
TRANSCRIPT
FUNCTIONS
Relation – a set of ( x , y ) points
Function – a set of ( x , y ) points where there is only
one output for each specific input
– x can not be paired with more than one y
** just make sure x doesn’t repeat itself
Ways to represent functions :
1. A set of ( x , y ) coordinates
2. A mapping
3. An equation
FUNCTIONS
Relation – a set of ( x , y ) points
Function – a set of ( x , y ) points where there is only
one output for each specific input
– x can not be paired with more than one y
** just make sure x doesn’t repeat itself
Ways to represent functions :
1. A set of ( x , y ) coordinates
2. A mapping
3. An equation
The DOMAIN of a function are its x values, its RANGE are the y values
This relation represents a true function, notice that the x coordinate never repeats…
( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )
This relation represents a true function, notice that the x coordinate never repeats…
( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )
Lets look at a mapping. Mapping has the x’s on one side and the y’s on the other side. You draw arrows showing which x is associated with which y.
This relation represents a true function, notice that the x coordinate never repeats…
( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )
Lets look at a mapping. Mapping has the x’s on one side and the y’s on the other side. You draw arrows showing which x is associated with which y.
Let’s use the ( x , y ) pairs from the above example:
x y
1
2
-5
0
2
- 3
4
3
This relation represents a true function, notice that the x coordinate never repeats…
( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )
Lets look at a mapping. Mapping has the x’s on one side and the y’s on the other side. You draw arrows showing which x is associated with which y.
Let’s use the ( x , y ) pairs from the above example:
x y
1
2
-5
0
2
- 3
4
3
This relation represents a true function, notice that the x coordinate never repeats…
( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )
Lets look at a mapping. Mapping has the x’s on one side and the y’s on the other side. You draw arrows showing which x is associated with which y.
Let’s use the ( x , y ) pairs from the above example:
x y
1
2
-5
0
2
- 3
4
3
This relation represents a true function, notice that the x coordinate never repeats…
( 1 , 3 ) , ( 2 , 4 ) , ( - 5 , - 3 ) , ( 0 , 2 )
Lets look at a mapping. Mapping has the x’s on one side and the y’s on the other side. You draw arrows showing which x is associated with each y.
Let’s use the ( x , y ) pairs from the above example:
x y
1
2
-5
0
2
- 3
4
3
x y
1
2
-1
-2
1
4
Does the mapping show a true function ?
x y
1
2
-1
-2
1
4
Does the mapping show a true function ?
YES !!! Each x has only one y in the mapping
** its acceptable for y to have more than one match
x y
2
-2
1
-3
4
Does the mapping show a true function ?
x y
2
-2
1
-3
4
Does the mapping show a true function ?
NO !!! Notice that 2 has two matches.
x y
2
-2
1
-3
4
Does the mapping show a true function ?
NO !!! Notice that 2 has two matches.
If we showed the mapping as coordinates, you see that x repeats.
( 2 , 1 ) , ( 2 , 4 ) , ( - 2 , - 3 )
FUNCTIONS :
There is a special notation to show a function.
)(xf
FUNCTIONS :
There is a special notation to show a function.
)(xf - Read “f of x “
- There is a function f that has x as its variable
- it’s a different way of saying ”y”
- coordinate is ( x , f(x) )
FUNCTIONS :
There is a special notation to show a function.
53)( xxf
The rule of the function
FUNCTIONS :
There is a special notation to show a function.
53)( xxf
We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)
FUNCTIONS :
There is a special notation to show a function.
53)( xxf
We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)
When substituting a value for x, you show it in the f(x) notation…lets use x = 1
x f(x)
FUNCTIONS :
There is a special notation to show a function.
53)( xxf
We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)
When substituting a value for x, you show it in the f(x) notation…lets use x = 1
x f(x)
1
8)1(
5)1(3)1(
53)(
f
f
xxf
8
FUNCTIONS :
There is a special notation to show a function.
53)( xxf
We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)
When substituting a value for x, you show it in the f(x) notation…lets use x = 2
x f(x)
1
11)2(
5)2(3)2(
53)(
f
f
xxf
8
2 11
FUNCTIONS :
There is a special notation to show a function.
53)( xxf
We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)
When substituting a value for x, you show it in the f(x) notation…lets use x = 3
x f(x)
1
14)3(
5)3(3)3(
53)(
f
f
xxf
8
2 11
3 14
FUNCTIONS :
There is a special notation to show a function.
53)( xxf
We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)
You could keep going here, 3 points is enough for a linear function.
x f(x)
1 8
2 11
3 14
FUNCTIONS :
There is a special notation to show a function.
53)( xxf
We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)
Does this set of points satisfy a true function ?
x f(x)
1 8
2 11
3 14
FUNCTIONS :
There is a special notation to show a function.
53)( xxf
We can generate some ( x , f(x) ) points by picking some values for x, plugging them into the rule and solving for f(x)
Does this set of points satisfy a true function ?
YES…
x f(x)
1 8
2 11
3 14
FUNCTIONS :
62)2()2(
12)1()1(
22)0()0(
32)1()1(
102)2()2(
2)(
3
3
3
3
3
3
f
f
f
f
f
xxf
When generating points for quadratics, cubics, etc, it is a good idea to get 5 – 8 points ( + / - ) to show your relation and to eventually graph your function…
Below is the work to generate the ( x , f(x) ) coordinates for the given function…
x f(x)
-2
-1
0
1
2
-10
-3
-2
-1
6
FUNCTIONS :
When evaluating functions, sometimes algebraic expressions are used…
4)( 2 xxf
Complete the table below…
x )(xf
FUNCTIONS :
When evaluating functions, sometimes algebraic expressions are used…
4)( 2 xxf
Complete the table below…
4
x )(xf
FUNCTIONS :
When evaluating functions, sometimes algebraic expressions are used…
4)( 2 xxf
Complete the table below…
4 12
x )(xf 124
4164
444
42
2
f
f
f
xxf
FUNCTIONS :
When evaluating functions, sometimes algebraic expressions are used…
4)( 2 xxf
Complete the table below…
4 12
a
x )(xf
FUNCTIONS :
When evaluating functions, sometimes algebraic expressions are used…
4)( 2 xxf
Complete the table below…
4 12
a a2 – 4
x )(xf
4
4
4
2
2
2
aaf
aaf
xxf
FUNCTIONS :
When evaluating functions, sometimes algebraic expressions are used…
4)( 2 xxf
Complete the table below…
4 12
a a2 – 4
x )(xf
3m
FUNCTIONS :
When evaluating functions, sometimes algebraic expressions are used…
4)( 2 xxf
Complete the table below…
4 12
a a2 – 4
x )(xf
3m
563
4963
433
4
2
2
2
2
mmmf
mmmf
mmf
xxf
562 mm
FUNCTIONS :
When evaluating functions, sometimes algebraic expressions are used…
4)( 2 xxf
Complete the table below…
4 12
a a2 – 4
2x + h
x )(xf
3m
hxh
hxh
h
hxh
h
xhxhx
h
xhxhx
h
xhx
h
xfhxf
xxf
2
22
442
442
44
4
2
222
222
22
2
562 mm
h
xfhxf