Functions and Graphs€¦ · Functions and Graphs Domain Range Domain Range 0 5 0 5 1 6 1 6 2 9 2 9...
Transcript of Functions and Graphs€¦ · Functions and Graphs Domain Range Domain Range 0 5 0 5 1 6 1 6 2 9 2 9...
Functions and Graphs
Functions and Graphs
An ordered pair is two numbers written
in a certain order, usually written in
parentheses. For example: (3, 2) is an
ordered pair with three as the first
number and two as the second number.
We often use an ordered pair to
represent coordinates on the Cartesian
Plane. The first number representing
the horizontal distance from the origin
(0,0) and the second number
representing the vertical distance.
x (4,6)
4
6
Functions and Graphs
Quadrant I
Quadrant II Quadrant III
Quadrant IV
(pos, pos)
(pos, neg) (neg, neg)
(neg, pos)
The Cartesian Plane is divided into
quadrants.
They are numbered counterclockwise.
*(3,1)
* (3,-1) (-3,-1) *
(-3,1) *
Functions and Graphs
A set of ordered pairs is called a relation between two
variables. These relations of two variables may be
represented by graphs on the Cartesian Plane.
Example:
Graph {(-7,3),(-4,-2),(4,6),(4,3)}
* ,6)
* (-4,-2)
* (4,3)
* (4,6)
* (-7,3)
Functions and Graphs
Sometimes we have a rule to represent our relation:
Examples:
Y = 3x – 3
(x-3)2 + (y-5)2 = 4
Functions and Graphs
Given a relation as a rule, we
can sketch a graph by
generating a table of values.
(-2,7)
(-1,3)
(0,-1)
(1,-5)
(2,-9)
Functions and Graphs
A function is a relation between two sets, one called
the domain and the second called the range. In a
function, each member of the domain has exactly one
corresponding member in the range. When a function
is represented by ordered pairs (x,y), x represents
members of the domain, and y members of the range.
Domain Range
Functions and Graphs
Domain Range Domain Range
0 5 0 5
1 6 1 6
2 9 2 9
5
The relation on the left is a function. Each element of the domain has exactly one
corresponding element in the range. The relation on the right is not a function because zero
has two corresponding domain elements.
If you think of domain as input and range as output, a function has only one output for each
input.
Functions and Graphs
Find the domain and range of the function: {(-7,3),(-4,-
2),(4,6),(4,3)}
Domain {-7,-4,4,5}
Range {3,-2,6,3}
* 6)
* (-4,-2)
* (5,3)
* (4,6)
* (-7,3)
Functions and Graphs
Find the domain and range of the function shown:
Domain: < 6
Range: < 4
Note that open circles indicate
strictly < or >.
−3 ≤ 𝑦
−5 ≤ 𝑥
domai
n
R
A
N
G
E
Functions and Graphs
Find the domain and range of the function: y = X+1
Domain: All real numbers
Range: All real numbers
Functions and Graphs
Find the domain and range of the function: y =
Domain: X > 0
Range: All real numbers
𝑋
Functions and Graphs
Vertical Line Test If a vertical line crosses the path of a graph in more than on place, it is
not a graph of a function.
Not a function. Fails test No evidence this is not a function
Functions and Graphs
Vertical Line Test If a vertical line crosses the path of a graph in more than on place, it is
not a graph of a function.
Not a function. Fails test No evidence this is not a function
Functions and Graphs
Equations of Functions We can represent functions by equations by defining one
variable in terms of another. For instance y = 7x. We can
think of x as an input value and y as an output. This leads to
the model of a function machine:
x
y
In our example if 2 goes in, 14 comes out.
If two is in our domain, then 14 is generated
as value in our range.
Functions and Graphs
Functional Notation This model helps in understanding functional notation. With functional
notation, instead of writing y = 2x, we would write f(x) = 2x. Using this
notation, f(2) = 2(2) = 14. There is no mathematical operation
occurring on the f(x) side of the equation, except to show we
substituted 2 in as our input. f(x) is read as “f of x”, “f at x” or “the
value of f at x.”
x
f(x)
In our example if 2 goes in, 14 comes out.
If two is in our domain, then 14 is generated
as value in our range.
Functions and Graphs
Functional Notation Try these:
a) 𝑓 𝑥 = 3𝑥 − 1; 𝑓𝑖𝑛𝑑 𝑓(−2)
b) 𝑓 𝑥 = 5𝑥2 − 2𝑥; 𝑓𝑖𝑛𝑑 𝑓(3)
c) 𝑓 𝑥 = 2𝑥 − 5; 𝑓𝑖𝑛𝑑 𝑓 𝑔 − 4
Functions and Graphs
Functional Notation Try these:
a) 𝑓 𝑥 = 3𝑥 − 1; 𝑓𝑖𝑛𝑑 𝑓(−2) f(-2) = 3(-2) – 1 = -7
b) 𝑓 𝑥 = 5𝑥2 − 2𝑥; 𝑓𝑖𝑛𝑑 𝑓(3) f(3) = 5(32) – 2(3) = 5*9 – 6 = 39
c) 𝑓 𝑥 = 2𝑥 − 5; 𝑓𝑖𝑛𝑑 𝑓 𝑔 − 4 f(g-4) = 2(g - 4) – 5 = 2g – 8 -5
= 2g - 13