Functions

An Overview

Functions

• A function is a procedure for assigning a single output to any acceptable input.

• Functions can be written as formulas, tables, graphs, or verbal descriptions.

• Functions are used to describe the relationship between two or more quantities.

• As the quantities change with respect to each other (cause), they create predictable behaviors (effect) that can be described.

Basic & Elementary Functions• Basic functions are power functions, x k

(where k is any real number), the trigonometric functions sin(x) and cos(x), and the natural exponential e x and logarithm functions ln (x)

• An elementary function is a combination of any of these functions through addition, subtraction, multiplication, division, and composition.

Graphs

• The graph of a function f is the set of points (x, y) that satisfy the equation y = f(x).

• Graphs may or may not have a formula to represent f(x).

Function Notation

• Function notation is used to succinctly state the formula for a function and related points.

• The formula y = 2x written in function notation would appear as f(x) = 2x.

• Any letter can be chosen as the name of a function, although the letters f, g, and h are the most common.

Function Notation

• A particular point on the function f(x) = 2x, such as x = 4, is written as f(4) = 8. The corresponding ordered pair notation for this point is (4, 8).

• The statement y = f(x) is used to indicate that the equation y is also a function in terms of x.

Types of Functions

• Algebraic functions are combinations of power functions using only addition, subtraction, multiplication and division.

• Polynomial functions are algebraic functions with the exponents restricted to whole numbers.

• A rational function is the quotient of two polynomial functions.

Types of Functions

• A transcendental function is any function which is not algebraic.

• Exponential, logarithm, and trigonometric functions are all transcendental.

Relations

• Any set of ordered pairs

• Can be represented as a list: {(0, 1); (3, -6); (-7, 2); (3, 5); (0, 2)} inequality: y > x equation: x2 + y2 = 9 function: y = x2

Domain and Range

• Domain: The set of first entries of the ordered

pairs In a function, the x-values or inputs

• Range: The set of second entries of the

ordered pairs In a function, the y-values or outputs

Function• A relation that assigns only one range

value to any given domain value• The formula cannot contain an

inequality symbol nor an exponent on the y variable

• The graph of a function will pass the vertical line test A vertical line in the xy-plane will intersect

the graph of a function in at most one point.

Function Models

• Functions can model real-life situations in which the values of 2 or more variables are related

• Many models exist where one variable depends on another x is the independent variable y is the dependent variable

• y is a function of x )(xfy

Types of Functions

• Algebraic Polynomial Rational Root

• Piecewise Defined Absolute Value Greatest Integer

• Transcendental Trigonometric Exponential Logarithmic

• Combinations Sums/Differences Products/Quotients Composite

Function Notation

is the same as

The input variableaka, the “dummy” variable

can be replaced by a number, another variable, or an expression

123)( 2 xxxf

123)( 2 wwwf

12223)2( 2 f9)2( f

1)4(2)4(3)4( 2 xxxf

182)168(3)4( 2 xxxxf

18248243)4( 2 xxxxf

41223)4( 2 xxxf

is the same as (2, 9)

Identifying Domain & Range

• y = x Domain: (-∞, ∞) Range: (-∞, ∞)

• y = x2

Domain: (-∞, ∞) Range: [0, ∞)

• y = c(x – a)2 + b, c < 0

Domain: (-∞, ∞) Range: (-∞, b]

• y = x3

Domain: (-∞, ∞)

Range: (-∞, ∞)

• y = c(x – a)2 + b, c > 0

Domain: (-∞, ∞)

Range: [b, ∞)

• y = x4

Domain: (-∞, ∞)

Range: [0, ∞)

Algebraic Approach for Range

• For quadratic functions, “complete the square” to get it in vertex form

852 2 xxy

xxy 528 2 xxy

252

21 8

225

21

2522

25

21

21 8 xxy

221 25.15625.14 xy

221 25.14375.2 xy

4375.225.1 221 xy

875.425.12 2 xy

Range: [4.875, ∞)

Identifying Domain & Range

• Domain:

[0, ∞) Range:

[0, ∞)

• Domain:

[a, ∞) Range:

[b, ∞)

xy baxy

Algebraic Approach for Domain

• For root functions, set the radicand greater than or equal to zero and solve for x

43)( xxf

043 x

43 x

3

4x

Domain: [4/3, ∞)

Identifying Domain & Range

•

Domain: (-∞, 0) (0, ∞)

Range:(-∞, 0) (0, ∞)

•

Domain:(-∞, a) (a, ∞)

Range:(-∞, b) (b, ∞)

xy

1 b

axy

1

Advice

• Unless otherwise stated, the domain of a relation is taken to be the largest set of real x-values for which there are corresponding real y-values

• Know the shapes of the basic graphs as well as their domains and ranges

Symmetry of Graphs

• About the y-axis If (x, y) is on the graph, so is (-x, y) Even Functions: f(-x) = f(x)

• About the x-axis If (x, y) is on the graph, so is (x, -y) Not a function

• About the origin If (x, y) is on the graph, so is (-x, -y) Odd Functions: f(-x) = -f(x)

Examples

• f(x) = x2

f(-x) = (-x)2 = x2

f(-x) = f(x) even

• f(x) = x2 + 1 f(-x) = (-x)2 + 1 =

x2 + 1 f(-x) = f(x) even

• f(x) = x f(-x) = -x f(-x) = -f(x) odd

• f(x) = x + 1 f(-x) = -x+1 = 1-x f(-x) ≠ f(x) and

f(-x) ≠ -f(x) neither

Piecewise Defined Functions

• Functions which have different formulas applied to different parts of their domains

• Example

•

• Graphing on calculator – special technique

1

10

0

,1

,x

x,-

)( 2

x

x

x

xfy

Absolute Value Function

• As a piecewise defined function:

• MATH NUM 1:abs(

0

0

,x

x,-)(

x

xxxf

0

0

,2x

,2x-2)2(

x

xxxf

Example

• Draw a complete graph

• Evaluate algebraically

31)( xxxf

22

31

31)(

03 and 01 When

x

xx

xxxf

xx

4

31

31)(

03 and 01 When

xx

xxxf

xx

22

31)(

03 and 01 When

x

xxxf

xx

Integer-Valued Functions

• Greatest Integer Functionaka, Integer Floor Function Greatest integer less than or equal to xMATH NUM 5:int(x)called a step functiongraph in dot modeNote y-values for negative x-values

xxxf )(

Integer-Valued Functions

xxf )(

• Integer Ceiling Function Round x up to the nearest

integerLeast integer greater than

or equal to x

fggfgfgfgf / ,/ , , ,

• The sum, difference, or product of two functions is also a function Domain is the intersection of the

domains of f and g

• The quotient of two functions is also a function Domain is the intersection of the

domains of f and g AND excludes the roots of the denominator

function

Composition of Functions

• Output of 1 function becomes the input to another function

• Notations

• Domain of f g consists of the range of g which is also in the domain of f

))(( xgf

)(xgf

Example

7)( xxf 2)( xxg

7)(

7)()(

7)(

2

xxgf

xgxgf

xxf

3)2(

72)2(

7)2()2(2

gf

gf

ggf

Example

7)( xxf 2)( xxg

2

2

2

7)(

)()(

)(

xxfg

xfxfg

xxg

25)(

72)(

)2()2(2

2

xfg

xfg

ffg