we saw yesterday that every relationship between x- and y-values represent a relation. that means...
TRANSCRIPT
We saw yesterday that every relationship between x- and y-values represent a relation.
That means every graph on a coordinate grid represents a relation
How can we write thisrelation down?
As ordered pairs,a table, or a mappingdiagram.
{(1,5),(3,5),(4,2),(6,-2)} Is it a function?
How can we writethis relation down?
We cannot write every point on this graph down – there is always another in between so the only way is to write an equation.
y = 2x + 4 Is this a function?
D: {1,3,4,6} R: {-2,2,5}
D: x can be anything
R: y can be anything
A better way is to use set notation
To use the symbols of algebra, we could write the domain as
Rxx :
Does that look like a foreign language?
Let’s translate:
The curly braces
just tell us we have a set of numbers.
The x reminds us
that our set contains x-values.
x
The colon says,
such that
:x
: xx
The symbol that looks like an e
(or a c sticking its tongue out)says, belongs to
or is an element of. . .
And the cursive, or script,
R
R xx:
is short for the set of real numbers.
R, the set
So we read it,
“The set of x
:
such thatx belongs to
x Rx
of real numbers.”
“The set of y, such that
y belongs to R,
the set of real numbers.”
Read this:
Ryy :
the domain and range can be any real number.
It is not always true that
Sometimes mathematicianswant to study a function over
a limited domain.
What do you think of the domain?
What about the range?
}:{ Rxx
}2,:{ yRyy
What do you think of the domain?
What about the range?
Function or not?
}61,:{ xRxx
}61,:{ yRyy
What do you think of the domain?
What about the range?
Function or not?
}51,:{ xRxx
}32,:{ yZyy
HW WorksheetDomain and Range
• When we know that a relation is a function, the “y” in the equation can
be replaced with f(x).• f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’.
• The ‘f’ names the function, the ‘x’ tells the variable that is being used.
Since the equation y = x - 2 represents a function, we can also write it as f(x) = x - 2.
Find f(4):f(4) = 4 - 2f(4) = 2
If g(s) = 2s + 3, find g(-2).g(-2) = 2(-2) + 3
=-4 + 3 = -1
g(-2) = -1
If h(x) = x2 - x + 7, find h(2c).h(2c) = (2c)2 – (2c) + 7
= 4c2 - 2c + 7
If f(k) = k2 - 3, find f(a - 1) f(a - 1)=(a - 1)2 - 3 (Remember FOIL?!)
=(a-1)(a-1) - 3 = a2 - a - a + 1 - 3 = a2 - 2a - 2
pg 635 #2, 4, 6, 8 (no sketch)
Solve the equation for y.
Substitute any value for x and find how many answers it produces for y.
One: function More than one: not
a function
2x + 4y = 8 y = -0.5x + 2 This equation
produces one output for every input so it is a function
xy
xy
4
16 22
This equation will produce two outputs for every input and is therefore not a function
If and find:
2
1)(
x
xxf xxg 4)(
2
174
2
)2(41
42
1
2
x
xx
x
xxx
xx
x
gf
If and find:
2
1)(
x
xxf xxg 4)(
xx
xxx
x
xx
x
gf
84
14
1
2
1
42
1
/
2
If and find:
2
1)(
x
xxf xxg 4)(
2
174
2
)2(41
42
1
2
x
xx
x
xxx
xx
x
gf
Worksheet
Finding and Graphing