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


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


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) *


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)


Sometimes we have a rule to represent our relation:

Examples:

Y = 3x – 3

(x-3)2 + (y-5)2 = 4


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)


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


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.


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)


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


Find the domain and range of the function: y = X+1

Domain: All real numbers

Range: All real numbers


Find the domain and range of the function: y =

Domain: X > 0

Range: All real numbers

𝑋


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


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.


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.


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


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

functions and graphs€¦ · functions and graphs domain range domain range 0 5 0 5 1 6 1 6 2 9 2 9...

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