2.3 – Functions and Relations
Relations, Domain and Range
Defn: A relation is a set of ordered pairs.
A  0,0, 1,1, 1,1, 4,2, 4,2
Domain: The values of the 1st component of the ordered pair.
domain A : 0, 1, 4
Range: The values of the 2nd component of the ordered pair.
range  A :  2, 1, 0, 1, 2
2.3 – Functions and Relations
Relations, Domain and Range
State the domain and range of each relation.
x
y
x
y
1
3
4
2
2
5
-3
8
-4
6
6
1
1
4
-1
9
3
3
5
6
domain :  4, 1, 2, 3
range : 3, 4, 5, 6
domain :  3, 1, 4, 5, 6
range : 1, 2, 6, 8, 9
2.3 – Functions and Relations
Functions
Defn: A function is a relation where every x value has one and only one
value of y assigned to it.
State whether or not the following relations could be a function or not.
x
y
x
y
x
y
4
2
1
3
2
3
-3
8
2
5
5
7
6
1
-4
6
3
8
-1
9
1
4
-2
-5
5
6
3
3
8
7
function
not a function
function
2.3 – Functions and Relations
Functions
State whether or not the following mappings of two relations could be a
function or not.
x
y
x
y
Relation A is not a function.
Relation B is a function.
For x = 1, there are two y
values (3 and 4).
For each value of x, there is
one value of y.
2.3 – Functions and Relations
Functions
Graphs can be used to determine if a relation is a function.
Vertical Line Test
If a vertical line can be drawn so that it intersects a graph of an
equation more than once, then the equation is not a function.
2.3 – Functions and Relations
Functions
The Vertical Line Test
y
y  2x  3
x
y
0
-3
5
7
-2
-7
4
5
3
3
function
x
2.3 – Functions and Relations
Functions
The Vertical Line Test
y
x  y2
x
y
1
1
1
-1
4
2
4
-2
0
0
not a function
x
2.3 – Functions and Relations
Functions
Domain and Range from Graphs
y
Find the domain and
range of the function
graphed to the right.
Use interval
notation.
Domain
x
Domain: [–3, 4]
Range: [–4, 2]
Range
2.3 – Functions and Relations
Functions
Domain and Range from Graphs
Find the domain
and range of the
function graphed to
the right. Use
interval notation.
y
Range
x
Domain: (– , )
Range: [– 2, )
Domain
2.3 – Functions and Relations
Function Notation
Shorthand for stating that an equation is a function.
Defines the independent variable (usually x) and the
dependent variable (usually y).
y  3x  1
y  x   3x  1
y  yx   f x 
f x  3x  1
2.3 – Functions and Relations
Function Notation
Function notation also defines the value of x that is to be use
to calculate the corresponding value of y.
f x   2 x  5
find f(3).
f 3  23  5
f 3  1
3, 1
g(x) = x2 – 2x
find g(–3).
f(x) = 4x – 1
find f(2).
f(2) = 4(2) – 1
g(–3) = (-3)2 – 2(-3)
f(2) = 8 – 1
f(2) = 7
(2, 7)
g(–3) = 9 + 6
g(–3) = 15
(–3, 15)
2.3 – Functions and Relations
Function Notation
Given the graph of
the following function,
find each function
value by inspecting
the graph.
y
f(x)
f(5) = 7
●
f(4) = 3
f(5) = 1
f(6) = 6
●
●
●
x
2.4 – linear Equations in Two Variables and Linear Functions
Three Forms of an Equation of a Line
Slope-Intercept Form:
This form is useful for graphing, as the slope and the y-intercept are
readily visible.
Point-Slope Form:
The point-slope form allows you to use ANY point, together with the
slope, to form the equation of the line.
Standard Form:
Note: A, B, and C cannot be fractions or decimals.
2.4 – linear Equations in Two Variables and Linear Functions
Slope of a Line
y y2  y1

Slope formula = m 
x x2  x1
Y
Y
Y
X
X
X
Green slant line:
positive slope
Vertical line:
undefined or no slope
Horizontal line:
slope = 0
Red slant line:
positive slope
Δ𝑥 = 0
Δ𝑦 = 0
Linear Functions
Constant Functions
2.4 – linear Equations in Two Variables and Linear Functions
Slope of a line
y y2  y1

Slope formula = m 
x x2  x1
Find the slope of the line containing the following points:
7−4
𝑚=
−3 − 1
𝑚=
3
3
=−
−4
4
−3, 7 𝑎𝑛𝑑 1, 4
2.4 – linear Equations in Two Variables and Linear Functions
Slope-Intercept Form: 𝑦 = 𝑚𝑥 + 𝑏
𝑚: 𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒
𝑏: 𝑡ℎ𝑒 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 0, 𝑏
8
𝑦=
𝑥−6
11
5𝑥 + 2𝑦 = 7
8
𝑚=
11
𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 0, −6
2𝑦 = −5𝑥 + 7
5
7
𝑦=− 𝑥+
2
2
5
𝑚=−
2
7
𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 0,
2
2.4 – linear Equations in Two Variables and Linear Functions
The Average Rate of Change
If f is defined on the interval [x1, x2], then the average rate of change
of f on the interval [x1, x2] is the slope of the secant line containing
(x1, f(x1)) and (x2, f(x2)).
Average rate of change:
f ( x2 )  f ( x1 )
y y2  y1
m
m

x2  x1
x x2  x1
2.4 – linear Equations in Two Variables and Linear Functions
Determine the average rate of change
From (–1.5, 0) to (0, 2)
𝑓 0 − 𝑓 −1.5
𝑚=
0 − −1.5
2
𝑚=
1.5
2−0
𝑚=
0 − −1.5
4
𝑚=
3
From (0, 2) to ( 2.5, 1)
𝑓 2.5 − 𝑓 0
𝑚=
2.5 − 0
−1
𝑚=
2.5
2
𝑚=−
5
1−2
𝑚=
2.5 − 0
2.4 – linear Equations in Two Variables and Linear Functions
Solving Equations and Inequalities Graphically
𝑦 = 𝑥 + 5 𝑎𝑛𝑑 𝑦 = −3𝑥 − 3
•
𝑥 + 5 = −3𝑥 − 3
4𝑥 + 5 = −3
4𝑥 = −8
𝑥 = −2
𝑥 + 5 > −3𝑥 − 3
−2, ∞
𝑥 + 5 ≤ −3𝑥 − 3
−∞, −2
2.4 – linear Equations in Two Variables and Linear Functions
Solving Equations and Inequalities Graphically
𝑦 = −4𝑥 + 4 𝑎𝑛𝑑 𝑦 = 2𝑥 − 2
−4𝑥 + 4 = 2𝑥 − 2
−6𝑥 + 4 = −2
•
−6𝑥 = −6
𝑥=1
−4𝑥 + 4 ≥ 2𝑥 − 2
−∞, 1
−4𝑥 + 4 ≤ 2𝑥 − 2
1, ∞