Understanding Understanding FunctionsFunctions

The set of all the x-values is called the Domain of the function.

For each element x in the domain, the corresponding element y is called the image of x.

The set of all images of the elements of the domain is called the Range of the function.

A function is a rule or a correspondence that associates each x-value with exactly one y-value.

4 ways to describe a 4 ways to describe a functionfunction

Mapping DiagramMapping Diagram

Ordered pairs/Table of valuesOrdered pairs/Table of values

GraphGraph

Rule (equation)Rule (equation)

Example: M is the Mother FunctionExample: M is the Mother Function

Joe

Samantha

Anna

Ian

Chelsea

George

Laura

Julie

Hilary

Barbara

Sue

Humans Mothers

1. Function as a Mapping.1. Function as a Mapping.

M: Mother functionM: Mother function

Domain of M Domain of M {{Joe, Samantha, Anna, Ian, Chelsea, GeorgeJoe, Samantha, Anna, Ian, Chelsea, George}}

Range of M Range of M {{Laura, Julie, Hilary, BarbaraLaura, Julie, Hilary, Barbara}}

In function notation we can write:In function notation we can write:M(M(AnnaAnna) = ) = JulieJulie or or M(M(GeorgeGeorge) = ) = BarbaraBarbara

Also, if we are told Also, if we are told M(M(xx) = ) = Hilary,Hilary, That means that x must be = That means that x must be =

ChelseaChelsea

For the function f below , evaluate f at the indicated values and find the Domain and Range of f

1

2

3

4

5

6

7

10

11

12

13

14

15

16

f(1) f(2)

f(3) f(4)

f(5) f(6)

f(7)

Domain of f:

Range of f:

{1, 2, 3, 4, 5, 6, 7}{10, 12, 13,

15}

2. Function as a Set of Ordered 2. Function as a Set of Ordered PairsPairs

A A functionfunction is a set of ordered pairs is a set of ordered pairs with the property that no two with the property that no two ordered pairs have the same first ordered pairs have the same first component and different second component and different second components. components.

In other words, you can’t have two In other words, you can’t have two different y-values for the same x-different y-values for the same x-value.value.

For each x, there is one related y-value

h:{(-2,3), (1,3), (4,5), (10,5)}j:{(1,-2), (2,2), (3,1), (4,-2)}p:{(0,0), (1,1)}

What is h(1)? What is j(1)?

What is p(1)?

For what values is h(x) = 5?

The mother function M can also be The mother function M can also be written as ordered pairswritten as ordered pairs

MM = = {{(Joe, Laura), (Samantha, Laura), (Joe, Laura), (Samantha, Laura),

(Anna, Julie), (Ian, Julie), (Chelsea, Hillary),(Anna, Julie), (Ian, Julie), (Chelsea, Hillary),

(George, Barbara) (George, Barbara) }}

3. Function as a 3. Function as a GraphGraph

Another way to depict a function, is to Another way to depict a function, is to display the ordered pairs on a graph display the ordered pairs on a graph on the coordinate plane, with the x-on the coordinate plane, with the x-values along the horizontal axis, and values along the horizontal axis, and the y-values on the vertical axis. the y-values on the vertical axis.

f = {(-3, -1), (-2, -3), (-1, 2), (0, -1), (1, 3), (2, 4), (3, 5)} is graphed below.

Domain of f = {-3, -2, -1, 0, 1, 2, 3}Range of f = {-3, -1, 2, 3, 4, 5}

43210-1-2-3-4

5

4

3

2

1

0

-1

-2

-3

x

y

x

y

4. Function Defined by a Rule4. Function Defined by a Rule

Let f be a function, consisting of ordered pairs where the second element of the ordered pair is the square of the first element.

Some of the ordered pairs in f are(1, 1) (2, 4), (3, 9), (4, 16),…….

f is best defined by the rule f(x) = x²

Function NotationFunction Notation f(x)

Functions defined on infinite sets are denoted by algebraic rules.

Examples of functions defined on all Real numbers

f(x) = x² g(x) = 2x – 1 h(x) = x³

The symbol f(x) represents the y-value in the Range corresponding to the Domain value x.

The point (x, f(x)) belongs to the function f.

Evaluating functionsEvaluating functions

5)f(

f(0)

f(3)

12xf(x)

-

g(3)

g(6)

g(1)3x

15g(x)

Determine the function values (y-values) Determine the function values (y-values) for the given x-values. for the given x-values.

5

-1

-11

-7.5

5

Undefined

2

Undefined3

If x is in the denominator, or in a square root, there will be restrictions on the Domain.

Graph of a functionGraph of a function

E.g.: The graph of the function

f(x) = 2x – 1 is the graph of the equationy = 2x – 1, which is a line.

Each point on the line is (x, f(x))

The graph of the function f(x) is the set of points (x, y) in the plane that satisfies the relation y = f(x).

Domain and Range from a Graph

Remember: Domain is the set of all x-values. On a graph, it is represented by all the values from left to right.

Range is the set of all the y-values. On a graph, it is represented by all the values from bottom to top.

For Real numbers, we write the Domain and Range in interval notation. [ #, # ]

Domain and Range from a Graph

Domain: x [-4, +[

Range: y [-3, +[

4

0

-4

(-4, 2)

x

y

4-4

The Zero of a Function

The zero of a function is the place where the function hits the x-axis. It is the x-intercept.

2

0

-2

x

y

2-2

What is the zero of the function graphed at the right?

The y-intercept of a Function

The y-intercept of a function is the place where the function hits the y-axis.

What is the y-intercept of the function graphed at the right?

2

0

-2

x

y

2-2

CalculatingCalculating the zero and y-intercept the zero and y-intercept of a function.of a function.

Calculate the zero of a function by making Calculate the zero of a function by making the function equal to zero and solving for x.the function equal to zero and solving for x.

Calculate the y-intercept by finding f(0).Calculate the y-intercept by finding f(0).

Given f(x) = 2x + 10, find:

a) the zero b) the y-intercept.

f(x) = 2x + 10 = 0 2x = -

10x = -5

f(0) = 2(0) + 10 = 10

y = 10

Calculate the Calculate the y-intercepty-intercept of of g:g:

Calculate the Calculate the zeroszeros of g: of g:

Consider the function:Consider the function:

g(x) = x g(x) = x2 2 + 3x – 4+ 3x – 4

g(0) = (0)2 + 3(0) – 4 = -4

g(x) = x2 + 3x – 4 = 0

(x + 4)(x – 1) = 0

x = -4 or x = 1

Consider the function:Consider the function:

g(x) = x g(x) = x2 2 + 3x – 4+ 3x – 4

g(0) = -4

The zeros arex = -4 or x = 1

g

5

Sign of the functionSign of the function

A function is positive where the graph is A function is positive where the graph is above the x-axis.above the x-axis.

It’s negative where the graph is below.It’s negative where the graph is below.

x

y positive

negative

-3

The function is positive on the interval x [-3, 5]

The function is negative on the intervals x ]- , -3] [5, + [1

Intervals of Increase or DecreaseIntervals of Increase or Decrease

5 x

y

-3

1

We need to identify where the function is increasing or decreasing

Increasing: x ]-, 1]

Decreasing: x [1, +[

Determine the Domain, Range, y-int, zeros, signs and intervals of increase and decrease for the following graph.

4

0

-4

(2, 3)

(7, -2.5)

x

y

Determine the Domain, Range, y-int, zeros, signs and intervals of increase and decrease for the following graph.

x

4

0

-4

(2, 3)

(7, -2.5)

y

Domain: Range:

y-int:Zeros:

Positive:

Negative:

Increasing:

Decreasing:

Extrema (max/min):

Theorem Vertical Line Test

A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.

x

y

Not a function.

x

y

Function.

4

0

-4

(2, 3)

x

y

Is this a graph of a function?