The document discusses linear relations and functions. It defines relations and functions, and explains how to determine if a relation is a function based on whether the domain contains repeating x-values. It shows how to represent relations as ordered pairs, tables, mappings, and graphs. It introduces the vertical line test to determine if a graph represents a function. It also explains function notation and how to find the value of a function for a given input.

1. 2-1: Graphing Linear Relations
and Functions
Objectives:
• Understand, draw, and determine if a
relation is a function.
• Graph & write linear equations,
determine domain and range.
• Understand and calculate slope.

2. Relations & Functions
Relation: a set of ordered pairs
Domain: the set of x-coordinates
Range: the set of y-coordinates
When writing the domain and range, do not
repeat values.

3. Relations and Functions
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}

4. Relations and Functions
• Relations can be written in several
ways: ordered pairs, table, graph, or
mapping.
• We have already seen relations
represented as ordered pairs.

5. Table
{(3, 4), (7, 2), (0, -1), x y
(-2, 2), (-5, 0), (3, 3)} 3 4
7 2
0 -1
-2 2
-5 0
3 3

6. 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.

7. Mapping
{(2, -6), (1, 4), (2, 4), (0, 0), (1, -6), (3, 0)}
2
-6
1
4
0
0
3

8. Functions
• 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.

9. Functions
• Discrete functions consist of points
that are not connected.
• Continuous functions can be
graphed with a line or smooth
curve and contain an infinite
number of points.

10. 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.
{(4, 1), (5, 2), (8, 2), (9, 8)}
Yes, no x-coordinate is repeated.

11. 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.

12. Does the graph represent a function?
Name the domain and range.
Yes
x D: all reals
R: all reals
y
Yes
x D: all reals
R: y ≥ -6
y

13. Does the graph represent a function?
Name the domain and range.
No
x D: x ≥ 1/2
R: all reals
y
No
x D: all reals
R: all reals
y

14. Does the graph represent a function?
Name the domain and range.
Yes
x D: all reals
R: y ≥ -6
y
No
x D: x = 2
R: all reals
y

15. 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.

16. Value of a Function
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 - 2
f(4) = 2

17. Value of a Function
If g(s) = 2s + 3, find g(-2).
g(-2) = 2(-2) + 3
=-4 + 3
= -1
g(-2) = -1

18. Value of a Function
If h(x) = x2 - x + 7, find h(2c).
h(2c) = (2c)2 – (2c) + 7
= 4c2 - 2c + 7

19. Value of a Function
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