FunctionsFunctionsandand

RelationsRelations

Relations

Definition:A relation is a set of ordered pairs.

Functions

Definition:A function is a relation such that for each x, there is exactly one corresponding y.

Definitions:

Domain is the set of all possible values of x of a function (or a relation).Range is the set of all possible values of __ of a function (or a relation).

is is isis

a function.a function.not a function.not a function.

y

–2, –1, 1, 3, 4–3, –1, 1–2, 4

3, 1, –11, 3, –1, 4, –22, –1

Examples:1. {(1, –1), (2, 1), (3, 3), (4, 5)}2. {(–2, 3), (–1, 1), (1, –1), (3, 1), (4, 3)}3. {(–3, 1), (–1, 3), (–1, –1), (1, 4), (1, –2)}4. {(–2, 2), (–2, –1), (4, 2), (4, –1)}

What are the domain and range of the four examples above?1. Domain = {1, 2, 3, 4} Range = {–1, 1, 3, 5}2. Domain = { } Range = { }3. Domain = { } Range = { }4. Domain = { } Range = { }

y

xO 2 4 6

2

4

6

–6

–4

–2–6 –4 –2

y

xO 2 4 6

2

4

6

–6

–4

–2–6 –4 –2

y

xO 2 4 6

2

4

6

–6

–4

–2–6 –4 –2

y

xO 2 4 6

2

4

6

–6

–4

–2–6 –4 –2

y

xO 2 4 6

2

4

6

–6

–4

–2–6 –4 –2

y

xO 2 4 6

2

4

6

–6

–4

–2–6 –4 –2

y

xO 2 4 6

2

4

6

–6

–4

–2–6 –4 –2

y

xO 2 4 6

2

4

6

–6

–4

–2–6 –4 –2

{(1, –1), (2, 1), (3, 3), (4, 5)}{(–2, 3), (–1, 1),

(1, –1), (3, 1), (4, 3)}{(–3, 1), (–1, 3),

(–1, –1), (1, 4), (1, –2)}{(–2, 2), (–2, –1), (4, 2), (4, –1)}

The Vertical Line Test:

The graph of a relation is a function if and only if every vertical line drawn only intersects the curve in at most one point.

Function Function Not a Function Not a Function

Function Notation

Ex. 1: {(1, 2), (2, 4), (3, 6), (4, 8)} Ex. 2: {(0, 0), (1, 1), (2, 4), (3, 9)}

f(x) = x2But the most conventional way to denote a function (Ex. 2) is:

Function-Value EvaluationEx. 1: f(x) = x2 + 1 a) f(2) = 22 + 1 = 5 b) f(5) = 52 + 1 = 26 c) f(abc) = (abc)2 + 1 = a2b2c2 + 1

Ex. 2: g(t) = t2 – 2t + 3 a) g(–3) = (–3)2 – 2(–3) + 3 = 18 b) g(2n) = (2n)2 – 2(2n) + 3 = 4n2 – 4n + 3

Ex. 3: H(z) = 2z2 – 5 a) H(4) = 2(4)2 – 5 = 27 b) H(z + 1) = 2(z +1)2 – 5 = 2(z2 + 2z + 1) – 5= 2z2 + 4z – 3

Note: f(x) is read as “f of x”, and it

doesn’t mean f times x; if it does, the notation would have been fx.

X Y

12

3

4

24

6

8

xf: 2x2xx f

X Y

01

2

3

01

4

9

xf: x2

x2x f

Name of the functionName of the variable (i.e., the input)

Definition of the function; here, the definition is: no matter what the input is, the output will be the square of the input.