Functions

Functions

Mathematics 4

June 20, 2012

Mathematics 4 () Functions June 20, 2012 1 / 1

DefinitionsRelations

Relations

A set of ordered pairs (x, y) such that for each x-value, there correspondsat least one y-value.


DefinitionsFunction

Function

A set of ordered pairs (x, y) such that for each x-value, there correspondsexactly one y-value.

Function

A correspondence from a set X ⊆ R to a set Y ⊆ R where x ∈ X andy ∈ Y , and y is unique for a specific value of x.


DefinitionsOne-to-one Function


DefinitionsDomain and Range

Domain

The domain of a function is the set of all possible values of x(independent variable, abscissa) for a given relation or function.

Range

The range of a function is the set of all possible values of y (dependentvariable, ordinate) for a given relation or function.


ExamplesDefinitions of Functions, Domain and Range

Identify if the following relations are functions, and give the domainand range.

1 y = x2 + 6x+ 4

2 x2 + y2 = 1

3 y =1

x+ 1

4 y =1

x2 + 1




1 y = x2 + 6x+ 4

2 x2 + y2 = 1

3 y =1

x+ 1

4 y =1

x2 + 1




1 y = x2 + 6x+ 4

2 x2 + y2 = 1

3 y =1

x+ 1

4 y =1

x2 + 1




1 y = x2 + 6x+ 4

2 x2 + y2 = 1

3 y =1

x+ 1

4 y =1

x2 + 1


Homework 5Identify if the following relations are functions, and give the domain and range.

1 y =3

x+ 1

2 y =3x2 + 1

x2 + 2

3 y =√−x2 + 25

4 x2 + y2 = 100

5 y + 3 = (x+ 4)2

6 y =2

|x|


Function Notation

Given the equation y = 2x2 + 5

Using the set-builder notation and the definition of functions:

f = {(x, y)∣∣y = 2x2 + 5}

From this notation we can use the shorthand:

f(x) = 2x2 + 5


Function Notation



f = {(x, y)∣∣y = 2x2 + 5}


f(x) = 2x2 + 5


Function Notation



f = {(x, y)∣∣y = 2x2 + 5}


f(x) = 2x2 + 5


DefinitionsGraphs of Functions

The graph of a function

The graph of a function is the set of ALL POINTS in R2 for which(x, y) ∈ a given function.

Vertical Line Test

The graph of a function can be intersected by a vertical line in at mostone point.


DefinitionsGraphs of Functions

The graph of a function

The graph of a function is the set of ALL POINTS in R2 for which(x, y) ∈ a given function.

Vertical Line Test

The graph of a function can be intersected by a vertical line in at mostone point.


Example:Square root functions

Find the graph of the function f = {(x, y)∣∣y =

√4− x}:


Example:Square root functions

Find the graph of the function f = {(x, y)∣∣y =

√x− 1}:


Example:Absolute value functions

Find the graph of the function f = {(x, y) |y = |x− 3|}:


Homework 6Sketch the graph and determine domain and range for each function below.

1 f = {(x, y) | y =√16− x2}

2 g = {(x, y) | y = (x− 1)3}

3 h =

{(x, y) | y =

x2 − 4x+ 3

x− 1

}


Evaluating Functions


Assign values to the independent variable and simplifying.

Evaluate the following:

1f(x+ h)− f(x)

hif f(x) = 3x2 − 2x+ 4

2 f(−x) if f(x) = 3x4 − 2x2 + 7

3 g(−x) if g(x) = 3x5 − 4x3 − 9x

4 f(x2 − 1) if f(x) =1 + x

2x− 1






1f(x+ h)− f(x)

hif f(x) = 3x2 − 2x+ 4

2 f(−x) if f(x) = 3x4 − 2x2 + 7

3 g(−x) if g(x) = 3x5 − 4x3 − 9x

4 f(x2 − 1) if f(x) =1 + x

2x− 1






1f(x+ h)− f(x)

hif f(x) = 3x2 − 2x+ 4

2 f(−x) if f(x) = 3x4 − 2x2 + 7

3 g(−x) if g(x) = 3x5 − 4x3 − 9x

4 f(x2 − 1) if f(x) =1 + x

2x− 1






1f(x+ h)− f(x)

hif f(x) = 3x2 − 2x+ 4

2 f(−x) if f(x) = 3x4 − 2x2 + 7

3 g(−x) if g(x) = 3x5 − 4x3 − 9x

4 f(x2 − 1) if f(x) =1 + x

2x− 1


Operations on Functions

The following notations are indicate an operation between two functions:

(f + g)(x) = f(x) + g(x)(f − g)(x) = f(x)− g(x)

(f · g)(x) = f(x) · g(x)(f

g

)(x) =

f(x)

g(x)



Determine the result of the following function operations:

1 (f + g)(x) if f(x) =x+ 3

x+ 2and g(x) = x− 2

2

(f

g

)(x) if f(x) =

√x3 − x2 − 5x− 3 and g(x) =

√x− 3



Determine the result of the following function operations:

1 (f + g)(x) if f(x) =x+ 3

x+ 2and g(x) = x− 2

2

(f

g

)(x) if f(x) =

√x3 − x2 − 5x− 3 and g(x) =

√x− 3


Composition of FunctionsDefinition

Composition of Functions

Evaluating a function f(x) with another function g(x).

f (g(x)) = (f ◦ g)(x)



Evaluate the following composite functions:

1 f(x) =x+ 1

x− 1, g(x) =

1

x, find (f ◦ g) and (g ◦ f)

2 f(x) =√x2 − 1, g(x) =

√x− 1, find (f ◦ g)(x) and (g ◦ f)



Evaluate the following composite functions:

1 f(x) =x+ 1

x− 1, g(x) =

1

x, find (f ◦ g) and (g ◦ f)

2 f(x) =√x2 − 1, g(x) =

√x− 1, find (f ◦ g)(x) and (g ◦ f)


Odd and Even FunctionsDefinitions

Even Function

A function f is even if f(−x) = f(x).

Example

1 f(x) = 3x6 − 2x4 + 4x2 + 2

2 g(x) = |x|+ 2



Even Function

A function f is even if f(−x) = f(x).

Example

1 f(x) = 3x6 − 2x4 + 4x2 + 2

2 g(x) = |x|+ 2



Odd Function

A function f is odd if f(−x) = −f(x).

Example

1 f(x) = 2x5 − 4x3 + 5x

2 g(x) =1

x



Odd Function

A function f is odd if f(−x) = −f(x).

Example

1 f(x) = 2x5 − 4x3 + 5x

2 g(x) =1

x


Odd and Even FunctionsDetermine if the following functions are odd, even, or neither

(1) (2)



(3) (4)



(5) (6)



(7) (8)


Odd and Even FunctionsSymmetry properties

Even functions

The graph of even functions are symmetric with respect to the y-axis.

Odd functions

The graph of odd functions are symmetric with respect to the origin.


Odd and Even FunctionsSymmetry properties

Even functions

The graph of even functions are symmetric with respect to the y-axis.

Odd functions

The graph of odd functions are symmetric with respect to the origin.


Homework 7Determine if the function is odd/even/neither, then find the domain, range, and thegraph of the function.

1 f(x) =x

x2 − 4

2 (f + g)(x) if f(x) = x2 + 1 and g(x) = |x|

3 (f − g)(x) if f(x) = |x|+ 1 and g(x) = x2 − 2

4 (f · g) if f(x) = x and g(x) = x3


Functions

Education

x x andy y

function f x

f x gxf f

4h2 f x iff x

h f x1if f x

f x gxf gx

set x r

specic value of x