# Polynomial Functions

## Relations

A relation is a set of ordered pairs where each ordered pair is in the form (a, b).

- Eg: S = { (1, 1), (1, 2), (3, 4), (5, 6) }

- Note the use of curly brackets { } and commas between each item in the list

The domain is the set containing all first elements

The range is the set containing all second elements

- Eg domain of S = {1, 3, 5}
- Eg range of S = {1, 2, 4, 6}

#### Example 1

… … If $y = x^2, \text{ for } x \in \{ 1, 2, 3, 4 \}$, find the set of ordered pairs and hence state the domain and range.

**Solution**

… … When x = 1, y = 1 so (1, 1)

… … When x = 2, y = 4 so (2, 4)

… … When x = 3, y = 9 so (3, 9)

… … When x = 4, y = 16, so (4, 16)

… … So the relation is: { (1, 1), (2, 4), (3, 9) (4, 16) }

… … Domain: $x \in \{ 1, 2, 3, 4 \}$

… … Range: $y \in \{ 1, 4, 9, 16 \}$

## Functions

A function is a relation where there is only one second element related to each first element in the domain.

In the example below,

- S is not a function because there are two different second elements related to the value of 1.
- Eg: S = { (1, 1), (1, 2), (3, 4), (5, 6) }

- T is a function because each first element only has one second element related to it.
- Eg: T = { (1, 8), (2, 6), (4, 2), (7, 6) }

## Vertical Line Test

A graph can be identified as a function if there is no vertical line that can be drawn that would touch the graph more than once.

If all vertical lines that can be drawn touch the graph either once or not at all then the graph is a function.

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#### Examples:

An upright parabola y = x^{2}.

- passes the vertical line test
- so y = x
^{2}is a function

.

.

A sideways parabola x = y^{2}.

- fails the vertical line test
- so x = y
^{2}is not a function

.

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## One to One and Many to One Functions

A One to One Function has only one x value for each y value and only one y value for each x value.

Informally, it passes both the vertical line test and the horizontal line test.

A Many to One Function has several x values with the same y value but only one y value for each x value.

Informally, it passes the vertical line test but fails the horizontal line test.

## Function Notation

The formal definition of a function looks like this:

… … f: D → CD, f(x) = {rule}

… … where

- f is the name of the function. Standard names for functions are f, g, h
- D is the domain of the function, written in interval notation
- CD is the Co-Domain, which is the set of values from which the range will be selected by the rule
- The Co-Domain is almost always R (the set of Real numbers)

The arrow —> is read as "maps onto",

- ie the values of the domain are mapped onto (or related to) values from the co-domai

f(x) = {rule} is where you write the rule for this function.

Examples of formally defined functions

… … f: R —> R, f(x) = 2x + 3

for the function f(x)

- Domain is R, the set of all real numbers

… … g: [0, 10) —> R, g(x) = x^{2} + 2

for the function g(x)

- Domain is [0, 10) which could be written as 0 < x < 10

… … h: R^{+} —> R, h(x) = sin(x)

for the function h(x)

- Domain is R
^{+}, which is the set of all positive real numbers, or x > 0

## Maximal and Implied Domain

The maximal domain for a function is the largest possible domain for which that function can be defined.

For many functions the maximal domain will be R (the set of all real numbers)

But for the square root function shown below, the maximal domain is x > 0

… … $f(x) = \sqrt{x}$

Implied Domain: If the function is written without a domain specified then it is implied that the domain is the maximal domain.

Therefore when we write the square root function above, we are implying that the formal definition is:

… … $f : R^+ \cup \{ 0 \} \rightarrow R, f(x) = \sqrt{x}$

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