What is aFunction?

by Judy Ahrens ~ 2005, 2006Pellissippi State Technical

Community College

Definition of a Function

slide 1

2x 3 13x = 4

3x2

6x = 4

4x 5 21x = 4

x x –2x = 4

REMEMBER!

Ima

Function

A function is a special relationship between twosets of elements. When you choose one elementfrom the first set, there must be exactly oneelement in the second set which goes with it.

ONE IN

ONE OUT

The Vertical Line TestIt is easy to recognize a function from its graph.A graph represents a function if and only if no vertical line intersects the graph more than once.

slide 2

a function a function

a function

not a function

not a functionnot a function

Recognizing a Function

slide 3

–2 0 3 4

4 0 916

The first relationship below defines a function,

01

4

0 1–1 2

but the second does not. Why not??

0149

1

What about the third relationship?

The second relationship pairs “1” with both “1” and “–1”, so it is not a function.The third relationship defines a function; each first element is paired with exactly one second element. The second elements can be the same!

Remember! One input

one output

Functional Notation

Equations are frequently used to represent functions.

slide 4

“x” is the independent variable

“f(x)” is the dependent variable

We may let y = f(x) on the graph of a function.

If f(x) = 4x + 5, then

f(x) = 4x + 5 is written in functional notation. We read it as “f of x equals 4x plus 5”.

If f(x) = 3 – 6, then2x2 36x 3xIf f(x) = , then

f(0) = 4(0) + 5 = 52

f(-7) = 3(-7) – 6 = 141

2 36( ) 3( ) 2 4 02 2 4 2 0 f(2) =

Domain of a Function

slide 5

–2 0 3 4

4 0 916

The domains below are sets of individual numbers.

D: {–2,0,3,4}

0149

1

D: {0,1,4,9}

The domains of the functions below are intervals.

D: ( ,2]

{(–5,2), (0,1), (4,–9), (7,6)}

D: { 5,0,4,7}

D: ( , )

The set of all x-values (inputs) is the domain.

D: (0, )

Implied DomainPolynomial Functions

slide 6

The domain of all polynomial functions is the set of all real numbers, i.e. . Examples:( , )

f (x) 3

f (x) 3x 8

Constant function

Linear function

Quadratic function

Cubic function

2 4x 1g(x) x 3 2h(x) x 8x 5

A term is a product of a number and a nonnegative integer power of a variable, e.g. 4 32x , 3x, 7, xA polynomial function is the sum or difference of terms, e.g. 4 3f (x) 2x 3x 7 x

Implied DomainRational Functions

slide 7

The domain of a rational function is the set ofall real numbers except those which would make the denominator equal zero.

3xf (x)2x 8

2

7g(x)x x 6

Examples:

2x 8 0 x 4 2x x 6 0

A rational function is the quotient of two polynomial functions, e.g.

3xf (x)2x 8

D : {x | x 4}, i.e.( , 4) (4, )

(x 3)(x 2) 0 x 2, 3 D : {x | x 2, 3}, i.e.( , 2) ( 2, 3) (3, )

Implied DomainRadical Functions

slide 8

The domain of radical functions with odd indicesis the set of all real numbers, i.e. .( , )

Examples: 3f (x) 3x 9,

The domain of a radical function with even indexis the set of all real numbers except those which make the radicand negative. .

Examples:

3x 9 0 x 3

f (x) 3x 9 4g(x) 8 4x

D : {x | x 2},i,e, [3, )D : {x | x 3},

8 4x 0 x 2

i.e. ( , 2]

5g(x) 8 4x

Range of a Function

slide 9

These ranges are sets of individual numbers.

R: {0, 4, 9, 16} R: {1}

The ranges of these functions are intervals.

R: [0, )

R: { 9, 1 ,2}

R: ( , )

The set of all y-values (outputs) is the range.

R: ( , 7]

–2 0 3 4

4 0 916

0149

1{(–5,2), (0,1), (4,–9), (7,1)}

For Practice

slide 10

1. Is it a function? {(0,0), (1,1), (4, 2), (4,-2)}

2. If f(x) = 8 – 3x: find f(-4), 3. Find the domain and range: 4f (x) 3x 2x 1

4. Find the domain of each function: 3 2g(x) 6x 2, f (x) 4 2x,

5xg(x)

3x 6

5. Find D & R

1. No, 4 is paired with 2 different numbers 2. 20, 3. Both are ( , ) 4.

( , ), ( ,2], ( , 2) ( 2, )

6. Are these graphs of functions?

D: ( , 2], R: ( ,5] 5.

The End

f(0), f(b), f(2a-b)

8-6a+3b8, 8-3b,

yes6. No, it fails the VLT;