1. Functions
and
Relations

2. Relations
Functions
Definition:
Definition:
A relation is a set of ordered pairs.
A function is a relation such that for each x, there is exactly one corresponding y.
Examples:
{(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)} is
a function.
a function.
not a function.
not a function.
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). y
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 = { }
–2, –1, 1, 3, 4
3, 1, –1
–3, –1, 1
1, 3, –1, 4, –2
–2, 4
2, –1

3. The graph of a relation is a function if and only if
{(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)} y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2
Function
Function
Not a Function
Not a Function y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2 y x O 2 4 6 –6 –4 –2
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.

4. X Y X Y Function Notation Ex. 1: {(1, 2), (2, 4), (3, 6), (4, 8)}1 2 3 4 9 x f: 2x x f: x2 2x x f x2 x f
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.
But the most conventional way to denote a function (Ex. 2) is:
f(x) = x2
Name of the function
Name 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.
Function-Value Evaluation
Ex. 1: f(x) = x2 + 1 a) f(2) = = 5 b) f(5) = = 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) – = 2z2 + 4z – 3