Language of Functions

A mapping transforms one set of values (inputs) into another set of values (outputs)
Mappings can be:
- One-to-one
  - Each input gets mapped to exactly one unique output
  - No two inputs are mapped to the same output
  - For example: A mapping that cubes the input
- Many-to-one
  - Each input gets mapped to exactly one output
  - Multiple inputs can be mapped to the same output
  - For example: A mapping that squares the input
- One-to-many
  - An input can be mapped to more than one output
  - No two inputs are mapped to the same output
  - For example: A mapping that gives the numbers which when squared equal the input
- Many-to-many
  - An input can be mapped to more than one output
  - Multiple inputs can be mapped to the same output
  - For example: A mapping that gives the factors of the input

Language of Functions Notes Diagram 2

A function is a mapping between two sets of numbers where each input gets mapped to exactly one output
- The output does not need to be unique

One-to-one and many-to-one mappings are functions
A mapping is a function if its graph passes the vertical line test
- Any vertical line will intersect with the graph at most once

Language of Functions Notes Diagram 4

Functions are denoted using letters (such as $f, v, g,$ etc)
- A function is followed by a variable in a bracket
- This shows the input for the function
- The letter $f$ is used most commonly for functions and will be used for the remainder of this revision note
$f (x)$ represents an expression for the value of the function $f$ when evaluated for the variable $x$
Function notation gets rid of the need for words which makes it universal
- $f = 5$ when $x = 2$ can simply be written as $f (2) = 5$

The domain of a function is the set of values that are used as inputs
A domain should be stated with a function
- If a domain is not stated then it is assumed the domain is all the real values which would work as inputs for the function
- Domains are expressed in terms of the input
  - $x \leq 2$
The range of a function is the set of values that are given as outputs
- The range depends on the domain
- Ranges are expressed in terms of the output
  - $f (x) \geq 0$
To graph a function we use the inputs as the x-coordinates and the outputs as the y-coordinates
- $f (2) = 5$ corresponds to the coordinates (2, 5)
Graphing the function can help you visualise the range
Common sets of numbers have special symbols:
- $ℝ$ represents all the real numbers that can be placed on a number line
  - $x \in ℝ$ means $x$ is a real number
- $ℚ$ represents all the rational numbers $\frac{a}{b}$ where a and b are integers and b ≠ 0
- $ℤ$ represents all the integers (positive, negative and zero)
  - $ℤ^{+}$ represents positive integers
- $ℕ$ represents the natural numbers (0,1,2,3...)

2-3-1-sets-of-numbers-diagram

Piecewise functions are defined by different functions depending on which interval the input is in
- E.g. $f (x) = \{\begin{matrix} x + 1 \\ 2 x - 4 \\ x^{2} \end{matrix} \begin{array}{l} x \leq 5 \\ 5 < x < 10 \\ 10 \leq x \leq 20 \end{array}$
The region for the individual functions cannot overlap
To evaluate a piecewise function for a particular value $x = k$
- Find which interval includes $k$
- Substitute $x = k$ into the corresponding function
The function may or may not be continuous at the ends of the intervals
- In the example above the function is
  - continuous at $x = 5$ as $5 + 1 = 2 (5) - 4$
  - not continuous at $x = 10$ as $2 (10) - 4 \neq 10^{2}$

Questions may refer to "the largest possible domain"
- This would usually be $x \in ℝ$ unless $ℕ$ , $ℤ$ or $ℚ$ has already been stated
- There are usualy some exceptions
  - e.g. square roots; $x \geq 0$ for a function involving $\sqrt{x}$
  - e.g. reciprocal functions; $x \neq 2$ for a function with denominator $(x - 2)$

For the function $f (x) = x^{3} + 1, 2 \leq x \leq 10$ :

write down the value of

f (7)

2-2-1-ib-ai-sl-language-of-functions-a-we-solution

find the range of

f (x)

2-2-1-ib-ai-sl-language-of-functions-b-we-solution

Language of Functions (DP IB Maths: AA HL)