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Transcript
```Do Now
Complete the chart for each linear equation.
y=x-2
x
y = 2x + 3
y
x
4
2
-3
-2
0
0
2
1
-1
-1
y
Appendix C
Functions
Objective: Determine if a relation is a function,
and find the domain and range.
Relations
Relation: a set of ordered pairs.
{ ( -3, 2), (-1, 1 ), ( 0, 7), (2, 4), (4, 3)}
{( -2, 1), (-1, 2), ( 0, 3), (1, 4), (2, 5)}
Independent Variable
( x, y )
Dependent Variable
The value of y “depends” on the value of x.
Domain: the set of all x-coordinates, independent variable
Range: the set of all y-coordinates, dependent variable
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}
Functions
Function: a relation in which, for each value of
the first component of the ordered pairs, there
is exactly one value of the second component.
• A function is a relation in which the members of
the domain (x-values) DO NOT repeat.
• So, for every x-value there is only one y-value that
corresponds to it.
• y-values CAN be repeated.
Ways to Represent a Function
• Symbolic
x,y y  2x
or
y  2x
• Numeric
• Graphical
X
Y
1
2
5 10
-1 -2
3
6
• Verbal
The cost is twice the
original amount.
Does the relation represent a
function?
H = {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)}
No, 3 is repeated in the domain.
G = {(4, 1), (5, 2), (8, 2), (9, 8)}
Yes, no x-coordinate is repeated.
Finding Domain & Range
Give the domain & range of each relation. Is it a
function?
Example 1
Example 2
x
y
1
1
-1
1
5
1
3
1
{(3, -6), (1, 3), (-2, 4),
(0,3), (1, -2), (3, 0)}
Vertical Line Test
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.
Finding Domain & Range
Give the domain & range of each relation. Is it a
function?
Finding Domain & Range
Give the domain & range of each relation. Is it a
function?
...
.
y
x
Does the graph represent a function?
Name the domain and range.
x
Function: Yes
D: All real numbers
R: All real numbers
y
x
y
Function: Yes
D: All real numbers
R: y ≥ -6
Does the graph represent a function?
Name the domain and range.
x
Function: No
D: x ≥ 1/2
R: All real numbers
y
x
y
Function: No
D: All real numbers
R: All real numbers
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 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.
• The parenthesis DO NOT mean multiplication!
• f(x) is another name for y.
• Sometimes other letters such as g, h or capital letters
F, G and H are used to name functions.
Using Function Notation
Find the value of each function.
1. If g(s) = 2s + 3, find g(-2).
2. If h(x) = x2 - x + 7, find h(2).
3. If f(x) = -x2 + 5x – 3 find f(q)
Homework
Page 457-458
#’s 12, 13, 18, 19, 35, 36, 37, 46
```
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