relations relation: a set of ordered pairs domain: the set of x-coordinates, independent range: the...
TRANSCRIPT
Relations
Relation: a set of ordered pairs
Domain: the set of x-coordinates, independent
Range: the set of y-coordinates, dependent
When writing the domain and range, do not repeat values.
Relations
Given the relation:{(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)}
State the domain:D: {0,1, 2, 3}
State the range:R: {-6, 0, 4}
Relations
• Relations can be written in several ways: ordered pairs, table, graph, or mapping.
• We have already seen relations represented as ordered pairs.
Table
{(2, -6), (1, 4), (2, 4),
(0, 0), (1, -6), (3, 0)}
x y 2 -6 1 4 2 4 0 0 1 -6 3 0
The ordered pairs should line up right next
to each other
Graphing
{(2, -6), (1, 4), (2, 4),
(0, 0), (1, -6), (3, 0)}
Plot the ordered pairs
Mapping
• Create two ovals with the domain on the left and the range on the right.
• Elements are not repeated. • Connect elements of the domain with
the corresponding elements in the range by drawing an arrow.
Mapping
{(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)}
2
1
0
3
-6
4
0
Functions
• Functions are relations that have exactly One output (y), dependent variable, for every input, independent variable (x)
• the members of the domain (x-values) DO NOT repeat.
• y-values, the range, can be repeated.
Do the ordered pairs represent a function?
{(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)}
No, 3 is repeated in the domain. When you input a 3, you can get a 4 or 3 out.
{(4, 1), (5, 2), (8, 2), (9, 8), (-4,3), (0,0)}
Yes, no x-coordinate is repeated. For each x there is only 1 y that is output.
Graphs of a Function
Vertical Line Test:
If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function.
x
y
x
y
Does the graph represent a function? Name the domain and range.
Yes
D: all reals
R: all reals
Yes
D: all reals
R: y ≥ -4
x
y
x
y
Does the graph represent a function? Name the domain and range.
NoD: x ≥ 1R: all reals
NoD: all realsR: all reals
Does the graph represent a function? Name the domain and range.
Yes
D: all reals
R: y ≥ -6
No
D: x = 2
R: all reals
x
y
x
y
Function Notation
• When we know that a relation is a function, the “y” in the equation can be replaced with f(x).
• f(x) is pronounced ‘f’ of ‘x’.• f(x) is the dependent variable, (output)• The ‘f’ names the function, the ‘x’ tells the
independent variable that is
being used.
Function Notation
• f(x) is the output or dependent variable
• We can Evaluate a function when we have an input
• We can then find the output
Value of a Function
Since the equation y = 3x + 4 represents a function, we can also write it as f(x) = 3x + 4
Find f(2): f(2) = 3(2) + 4f(2) = 6 + 4
f(2) = 10
The valve of output when x
is 2
Value of a Function
If f(x) = 2x , find f(-3).
f(-3) = 2(-3)
=-6
f(-3) = -6
Value of a Function
If f(x) = x2 + 3, find f(-4).
f(-4) = (-4)2 + 3
f(-4) = 16 + 3
f(-4) = 19
Operations with functions
• (f+g)(x) means to add the rule part of functions f(x) plus g(x)
• (f-g)(x) means to subtract the rule part of functions f(x) minus g(x)
Operations with functions
• (f g)(x) means to multiply the rule part of functions f(x) times g(x)
• ( )(x) means to divide the rule part of functions f(x) divided by g(x)
g
f
Operations with functions
Let f(x) = and g(x) =
1. (f + g)(x) =
(f + g)(x) =
342 xx 3x
)34( 2 xx )3( x
2x 5x 6
Operations with functions
Let f(x) = and g(x) =
2. (f / g)(x) =
(f / g)(x) =
(f / g)(x) =
342 xx 3x
)34( 2 xx
)3( x
( 3)( 1)x x
)3( x
1x
Operations with functions
Let f(x) = and g(x) =
3. (f – g)(x) =
(f – g)(x) =
(f – g)(x) =
342 xx 3x
)34( 2 xx )3( x
2 4 3x x 3x
2 3x x
Operations with functions
Let f(x) = and g(x) =
3. (f * g)(x) =
(f * g)(x) =
(f * g)(x) =
342 xx 3x
)34( 2 xx )3( x
3x 23x 24x 12x 3x 9
3x 27x 15x 9