The document provides an overview of linear relations and functions. It defines relations as sets of ordered pairs and functions as relations where each x-value corresponds to only one y-value. It discusses representing relations as ordered pairs, tables, mappings, and graphs. Key aspects of functions covered include discrete vs continuous functions, the vertical line test, function notation such as f(x), and evaluating functions by finding values such as f(4) given f(x) = x - 2.

1. 2-1: Graphing Linear Relations
and Functions bytarungehlot
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