Introduction to Functions (Edexcel IGCSE Maths)

Spec point — 3.2A

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 $\mathrm{f} ( x ) =...$ or $f:x\rightarrow...$

  • These two different types of notation mean exactly the same thing

  • 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

  • $f(x)=3x--4$

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

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

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

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

    • You cannot simplify this output any further

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

  • For example, given the function $f(x)=2x+1$

    • $f(3)=2\times3+1=7$
    • $f(-4)=2\times(-4)+1=-7$
    • $f(a)=2a+1$

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

  • For example, given the function $f(x)=2x+1$

    • If $f(x)=15$, then form an equation by replacing $\mathrm{f} ( x )$ with $2x+1$
    • $2x+1=15$
    • Solving this equation gives an input of 7

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

  • $\mathrm{f} ( x ) = 15$ means an input of $x$ gives an output of 15
  • $\mathrm{f} ( 15 )$ 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 $\mathrm{f} ( x ) = x + 3$ where $x\geq3$ could be shown as:

Worked Example A function is defined as $\mathrm{f} ( x ) = 3 x^{2} - 2 x + 1$.

(a) Find $\mathrm{f} ( 7 )$.

Answer:

The input is $x=7$, so substitute 7 into the expression everywhere you see an $x$

$\mathrm{f} ( 7 ) = 3 ( 7 )^{2} - 2 ( 7 ) + 1$* *

Calculate

$\begin{gathered} \mathrm{f} ( 7 ) & = 3 ( 49 ) - 14 + 1 \\ = 147 - 14 + 1 \end{gathered}$

$f(7)=134$

(b) Find $\mathrm{f} ( x + 3 )$, giving your answer in the form $ax^{2}+bx+c$ where $a$, $b$ and $c$ are integers to be found.

Answer:

The input is $( x + 3 )$ so substitute $( x + 3 )$ into the expression everywhere you see an $x$ This is like replacing $x$ with $( x + 3 )$

$\mathrm{f} ( x + 3 ) = 3 ( x + 3 )^{2} - 2 ( x + 3 ) + 1$

Expand the brackets and simplify Use that $( x + 3 )^{2} = ( x + 3 ) ( x + 3 )$ Be careful with negative signs

$\begin{gathered} \mathrm{f} ( x + 3 ) & = 3 ( x^{2} + 6 x + 9 ) - 2 ( x + 3 ) + 1 \\ = 3 x^{2} + 18 x + 27 - 2 x - 6 + 1 \\ = 3 x^{2} + 16 x + 22 \end{gathered}$

$f(x+3)=3x^{2}+16x+22$

Final answer: (a = 3, b = 16, c = 22)

A second function is defined by $g:x\rightarrow3x--4$.

(c) Find the value of $x$ for which $g:x\rightarrow-16$.

Answer:

This notation is not saying substitute 16 into the function (that would be $g:16$) It says that an input $x$ is substituted into $g$ giving the output -16 (i.e. $g ( x ) = - 16$) To find the input, form an equation by replacing $g:x$ with $3x-4$

$\begin{gathered} 3 x - 4 & = - 16 \end{gathered}$

Solve the equation (for example, by adding 4 to both sides, then dividing by 3)

$\begin{gathered} 3 x - 4 & = - 16 \\ 3 x & = - 12 \\ x = - \frac{12}{3} \end{gathered}$

$x=-4$