Language of Functions (DP IB Analysis & Approaches (AA): SL): Revision Note

Language of functions

What is a mapping?

• 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

    • {1, 2, 3, ...} maps to {1, 8, 27, ...}

  • 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

    • {±1, ±2, ±3, ...} map to {1, 4, 16, ...}

  • 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

    • {1, 4, 16, ...} maps to {±1, ±2, ±3, ...}

  • 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

    • e.g. the factors of {2, 3} are {1, 2, 3}

    • one input, e.g. {2}, has many outputs, e.g. {1, 2}

    • one output, e.g. {1}, has many inputs, e.g. {2, 3}

What is a function?

• 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

• This means a function can be

  • one-to-one

  • or many-to-one

• A sketch of the function must pass the vertical line test

  • Any vertical line will intersect with the graph at most once

    • e.g. $y=x^2$ and any vertical line $x=k$ pass the test

What notation is used for functions?

• Functions are denoted using letters (such as $f,v,g,$ etc)

  • If $x$ is the input

    • then $f\left(x\right)$ is the output of the function $f$

• e.g. if$f=5$ when$x=2$, then $f\left(2\right)=5$

What are the domain and range of a function?

• The domain of a function is the set of all 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

  • Domains are expressed in terms of $x$

    • e.g. $x\le 2$

• The range of a function is the set of all outputs

  • The range depends on the domain

  • Ranges are expressed in terms of $f\left(x\right)$

    • e.g. $f\left(x\right)\ge 0$

• To graph a function we use the inputs as the $x$-coordinates and the outputs as the $y$-coordinates

  • $f\left(2\right)=5$ corresponds to the coordinates (2, 5)

What sets of numbers do I need to know?

• Common sets of numbers have special symbols:

  • $\mathrm{ℝ}$ represents all the real numbers that can be placed on a number line

    • $x\in \mathrm{ℝ}$ means $x$is a real number

  • $\mathrm{\mathbb{Q}}$ represents all the rational numbers $\frac{a}{b}$where $a$ and $b$ are integers and $b\ne 0$

  • $\mathrm{\mathbb{Z}}$ represents all the integers (positive, negative and zero)

    • ${\mathrm{\mathbb{Z}}}^{+}$ represents positive integers

  • $\mathrm{\mathbb{N}}$ represents the natural numbers (0,1,2,3...)

What are piecewise functions?

• Piecewise functions are defined by different functions depending on which interval the input is in

  • E.g. $f\left(x\right)=\{\begin{array}{c}x+1\\ 2x-4\\ x^2\end{array} \begin{array}{l}x\le 5\\ 5<x<10\\ 10\le x\le 20\end{array}$

    • so $f\left(1\right)=1+1=2$

    • whereas $f\left(11\right)={11}^2=121$

• The region for the individual functions cannot overlap

• 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 both sides match: $5+1=2\left(5\right)-4$

    • not continuous at $x=10$ as $2\left(10\right)-4\ne {10}^2$