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Advanced Algebra Notes
Section 2.1: Represent Relations & Functions
A relation is a mapping, or pairing, of
input and output values. (A set of ordered pairs)
The 1st number in an ordered pair (Input value) is
called the
domain or x-coordinate .
The 2nd number in an ordered pair (Output value)
is called the
range
or y-coordinate .
A relation shows how each
member of the domain is paired with each
member of the range.
A function
is a special relation in
which each element of the domain is paired
with exactly one element of the range. This
is what is called one-to-one
correspondence.
Example 1: Identify the domain and range of the
relation and tell whether the relation is a function.
Input
-3
-1
2
4
Domain = { -3, -1, 2, 4 }
Range = { -4, 2, 3 }
Output
2
3
-4
Example 2: Represent the relation as a graph
and a mapping diagram.
{(-5,4), (-1,0), (3,-1), (3,5)}
Is the relation a function?
-5
-1
3
4
0
-1
5
When you are trying to determine if the graph of
a relation is a function you will use the
vertical line test . If the vertical line
intersects the graph in two or more places then
the relation is not a function, but if it only
intersects in one place then it is a function.
Many functions can be described by an
equation in two variables , such as y= -2x+5.
The input variable(x) is called the independent variable
The output variable(y) is called the dependent variable
because it depends on the value of the input variable.
solution
An ordered pair (x, y) is a
of an
equation in 2 variables if substituting x and y in the
equation produces a true statement.
Ex: y=3x–5, (2, 1) is a solution because 1=3(2)-5 is true.
graph
The
of an equation in 2
variables is the set of all points (x, y) that
represents solutions of the equation. The graph of
an equation in 2 variables is a linear function
if the equation is of the form y = mx + b.
The equation y = 3x – 5 is an example of a
linear function. You can write a linear function
in function notation by replacing the variable y
with f(x) to get f(x) = mx+b.
We read f(x) as “the value of x” or simply “f of x”
Graph the equation/functions:
y=-2x+1
y= -2(0)+1=1 (0,1)
y= -2(1)+1=-1 (1,-1)
f(x)=2/3 x-4
f(x)=2/3(0)-4 =-4
(0,-4)
f(x)=2/3(3)-4=-2
(3,-2)
We will also classify a function as linear
or not then evaluate the function.
A function is linear if its graph is
a line
or if the equation has
the x variable raised to the
first
power and the xvariable is not in the denominator
or inside an exponent
Tell whether the function is linear and then
evaluate when x = -3
f ( x)  2 x  5
3
f(x) = 2(-3*-3*-3)+5
f(x) = 2(-27)+5
f(x) = -54+5
f(x) = -49
f ( x)  12  8x
f(x) = 12 – 8(-3)
f(x) = 12 – (-24)
f(x) = 36