Functions

Relation - A set of ordered pairs x and y x is the input and y is the output

Function - A relation in which each x(input) value has exactly one y value(output)

Example

The function f(x) = x2 + 1

x = 1 then f(1) = 12 + 1 = 2

x = 2 then f(2) = 22 + 1 = 5

and so on

Some other common letters used to represent functions are:g(x), h(x), t(x), s(x)

The Verticle Line Test

Sweep a vertical line across the graph of the function. If the line crosses the graph more than once it is not a function, only a relation.

Identify which of the following our functions, or if they are just relations.

Identify which of the following are functions, or if they are just relations.

Set of ordered pairs {(1, 2), (1, 5), (2, 6), (7, 8)}

Set of ordered pairs {(1, 5), (2, 5), (3, 6)}

x f(x)

2

4

6

8

12

1416

18

20

Operations on Functions

+ - * ÷

When Adding or Multipyling functions, order in which you put them in doesn't matter, this is called the Commutative Law.

When Subtracting or Dividing, order in which you put them in does matter because it can result in different answers.

Commutativity+ •

– /

Operations on Functions

=

=

=

=

=

=

=

=

= =

Composite Functions

Take the output of one function and use it as an input for another function

Example

(f g)(x) = f(g(x)) Means to find the output for the function of g(x) and use it as the input for the function f(x)°

f(x) = 2x2 + 1 g(x) = x

3

(f g)(x) = f(x3) = 2(x

3)2 + 1 = 2x

6 + 1°

Or using numbers

Find (f g)(x) when x = 3°

Example

f(x) = (x + 1)(x) h(x) = 2x

Find and expression in terms of x for

(h f)(x) , then calculate the output for x = 2

°

°

Calculate the output for

f(h(x)) when x = 2

Given the functions f and g such that

f = {(2, 6)(3, 7)(4, 7)}

g = {(6, 10), (7, 12)}

find

a) f(2)

b) g(7)

c) g(f(2))

d) 2g(f(4)) - f(3)

Example

f(x) = 3x + 2 g(x) = x2

find

a) f(g(x))

b) g(g(x))

c) g(2) f(3)

d) 2f(4) - f(1) 3g(2)

Assignment

Exercise 51

1-8, and 8