Introduction to Functions

Introduction to Functionsvideo

Introduction to functions

What is a function?

• A function is a combination of one or more mathematical operations that takes a set of numbers and changes them into another set of numbers

• The numbers being put into the function are often called the inputs

• The numbers coming out of the function are often called the outputs

• A function may be thought of as a mathematical “machine”

  • For example, for the function “double the number and add 1”, the two mathematical operations are "multiply by 2 (×2)" and "add 1 (+1)"  

    • Putting 3 in to the function would give 2 × 3 + 1 = 7

    • Putting -4 in would give 2 × (-4) + 1 = -7 

    • Putting $x$ in would give $2x+1$

What is function notation?

• A function, f, with input $x$ can be written as $\mathrm{f}\left(x\right)=...$

  • Letters other than f can be used

    • The letters g, h and j are common but any letter can be used

    • Typically, a new letter will be used to define a new function in a question

• For example, the function with the rule “triple the number and subtract 4” would be written

  • $\mathrm{f}\left(x\right)=3x–4$ 

• In such cases, "$x$" is the input and "$\mathrm{f}\left(x\right)$" is the output

• Sometimes functions don’t have names like f and are just written as y = …

  • E.g. $y=3x–4$

How does a function work?

• A function has an input $x$ and an output $\mathrm{f}\left(x\right)$

  • If the input is 2, then the output is $\mathrm{f}\left(2\right)$

  • If the input is $m$, then the output is $\mathrm{f}\left(m\right)$

  • If the input is $t+5$, then the output is $\mathrm{f}\left(t+5\right)$

    • You cannot simplify this output any further

• If the function is known, the output can be calculated

  • For example, given the function $\mathrm{f}\left(x\right)=2x+1$

    • $\mathrm{f}\left(3\right)=2\times 3+1=7$

    • $\mathrm{f}(-4)=2\times (-4)+1=-7$

    • $\mathrm{f}\left(a\right)=2a+1$

• If the output is known, an equation can be formed and solved to find the input

  • For example, given the function $\mathrm{f}\left(x\right)=2x+1$

    • If $\mathrm{f}\left(x\right)=15$, then form an equation by replacing $\mathrm{f}\left(x\right)$ with $2x+1$

    • $2x+1=15$

    • Solving this equation gives an input of 7

• Note that $\mathrm{f}\left(x\right)=15$ and $\mathrm{f}\left(15\right)$ are very different things:

  • $\mathrm{f}\left(x\right)=15$ means an input of $x$ gives an output of 15

  • $\mathrm{f}\left(15\right)$ means substitute the input 15 into the function

What is a mapping diagram?

• A mapping diagram shows a set of different inputs going into the function to become a set of different outputs

  • Transforming inputs into outputs is called mapping

• For example, a mapping diagram for the function $\mathrm{f}\left(x\right)=x+3$ where $x\ge 3$ could be shown as: