02.10polynimialfuns

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.

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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 = x2.

  • passes the vertical line test
  • so y = x2 is a function

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A sideways parabola x = y2.

  • fails the vertical line test
  • so x = y2 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.

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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.

02.1type2.gif

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) = x2 + 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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