Functions, Function Notation, and Composition of Functions

A Way to Describe a Relationship

Relation A relationship between sets of information. Typically between inputs and outputs.

Function A relation such that there is no more than one output for each input

We have worked with many mathematical objects. For instance: equations, rules, formulas, tables, graphs, etc. In mathematics, similar things can also be described by the following vocabulary.

4 Examples of Functions

X Y

-3 1

-1 0

0 4

5 7

7 3

X Y

10 2

15 -5

18 -5

20 1

7 -5

These are all functions

because every x value has only one possible y

value

Every one of these functions is a

relation.

3 Examples of Non-Functions

X Y

0 4

1 10

2 11

1 -3

5 3

Every one of these non-functions is a

relation.

Not a function since x=1 can

be either y=10 or y=-3

Not a function since x=-4

can be either y=7 or y=1

Not a function since multiple x values have multiple y

values

The Vertical Line Test

If a vertical line intersects a curve more than once, it is not a function.

Use the vertical line test to decide which graphs are functions. Make sure to circle the functions.

The Vertical Line Test

If a vertical line intersects a curve more than once, it is not a function.

Use the vertical line test to decide which graphs are functions. Make sure to circle the functions.

Function Notation: f(x)Equations that are functions are typically written in a

different form than “y =.” Below is an example of function notation:

The equation above is read:

f of x equals the square root of x.

The first letter, in this case f, is the name of the function machine and the value inside the parentheses is the input. The expression to the right of the equal sign shows what the machine does to the input.

f x xDoes not stand for “f times x”

It does stand for “plug a value for x into a formula f”

Example

If g(x) = 2x + 3, find g(5).

5g 2 5 3

5 10 3g

5 13g

You want x=5 since g(x) was

changed to g(5)

When evaluating, do not write g(x)!

You wanted to find g(5). So the

complete final answer includes

g(5) not g(x)

A Justification for Function Notation

A function is similar to a factory machine. For the machine below, when 25 is the input (raw product) to the machine below, the output (finished product) is 5.

If ,

find 25 .

f x x

f

OR

25

5

The new notation reduces the amount of writing needed to express this substitution and evaluation. For instance:

Which do you prefer to write?

If ,

evaulate if 25.

y x

y x

Solving v Evaluating

23If 3, complete the following:f x x

a. Evaluate 3f b. Find if 5x f x Substitute and Evaluate

The input (or x) is 3.

Solve for x

The output is -5.

23 3 3 2

35 3x

No equal sign Equal sign

2 3

1

238 x

12 x

Substituting a function or it’s value into another function.

Composition of Functions

g xff

g

First

(inside parentheses always first)

Second

f g xOR

Example 1

Let and . Find:

1gf 2 5g x x 2 3f x x

211 5g

4

1 5

44 2 3f

11

8 3

This is an equivalent way to write it (The book does not use this notation):

1 11f g

Substitute x=1 into g(x) first

Substitute the result into f(x) last

1f g 1gf

4

Example 2

Let and . Find:

f xg 2 5g x x 2 3f x x

2 3f x x

22 3 52 3g xx

24 12 9 5x x 2 3 2 3 5x x

24 12 4g f x x x

Substitute x into f(x) first

Substitute the result into g(x) last

24 12 4x x

24 12 9 5x x

f xg

2 3x