Language of Functions (Cambridge International IGCSE Further Maths)

Introduction to Functions

What is a mapping?

• A mapping takes an 'input' from one set of values to an 'output' in another

• Mappings can be

  • 'many-one' (many 'input' values map to one 'output' value)

  • 'one-one' (one 'input' value maps to one 'output' value)

    • You may also come across 'many-many' and 'one-many' functions

What is a function?

• A function is a mapping where every 'input' value maps to a single 'output'
• Therefore only many-one and one-one mappings are functions

What notation is used for functions?

• Functions are denoted by $\mathrm{f} ( x ) , \mathrm{g} ( x )$, etc

  • e.g. $\mathrm{f} ( x ) = x^{2} - 3 x + 2$
  • These would be pronounced as 'f of x', 'g of x', etc

• There is an alternative notation

  • e.g. $\mathrm{f} : x \rightarrow \text{from} \text{bar} x^{2} - 3 x + 2$
  • Which may be pronounced 'the function f maps x to x-squared minus three x plus two'

How does a function work?

• A function has an input$( x )$ and output$( \mathrm{f} ( x ) \text{or} y )$

• Whatever goes in the bracket (instead of $x$) with f, replaces the $x$ on the other side

  • This is the input

• If the input 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 $f(x)=15$, the equation $2x+1=15$ can be formed
    • Solving this equation gives an input of 7

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

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

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 $\mathrm{f} ( x + 3 )$.

The input is $x=x+3$ so substitute $x+3$ into the expression everywhere you see an $x$.

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

Expand the brackets and simplify.

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

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

A second function is defined $\mathrm{g} : x \rightarrow \text{from} \text{bar} 3 x - 4$.

c) Find the value of $x$ for which $\mathrm{g} : x \rightarrow \text{from} \text{bar} - 16$.

Form an equation by setting the function equal to -16.

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

Solve the equation by first adding 4 to both sides, then dividing by 3.

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

$x=-4$

Domain & Range

What is the domain of a function?

• The domain of a function is the set of values that are allowed to be the ‘input’

• A function is only fully defined once its domain has been stated

  • If a domain is not stated then it is assumed that the domain is the largest set of possible values

    • e.g. the largest set of possible values for the function $\mathrm{f} ( x ) = \sqrt{x}$ would be $x\geq0$

• Restrictions on a domain can turn many-one functions into one-one functions

What is the range of a function?

• The range of a function is the set of values of all possible ‘outputs’
• The type of values in the range depend on the domain

How do I find a range from a given domain?

• The domain of a function is the set of values that are used as inputs

• The range of a function is the set of values that are given as outputs

• Finding the range of a function involves determining all possible output values from a given domain

  • This may need to be done by calculating each output value individually by **applying the function **to each input value
  • Or by considering the** shape** or pattern of the function

• 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

  • For example the range of the function $\mathrm{f} ( x ) = x^{2}$ for a domain of all real values of $x$ will be $\mathrm{f} ( x ) \ge 0$as the *y-*coordinates on the graph are all greater than or equal to zero

Worked Example The many-one function, $f(x)$, is given by

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

for all values of $x$.

a) State the range of $\mathrm{f} ( x )$.

The 'output' from the function $f$ is a squared value and so will be positive, or zero.

$f(x)\geq0$

b) The domain of $\mathrm{f} ( x )$ is changed to $x>5$. Write down the changed range of $\mathrm{f} ( x )$.

As $x>5$, $\mathrm{f} ( x ) > ( 5 - 3 )^{2}$

$\therefore f(x)>4$

The Modulus Function

What is the modulus function?

• The modulus function makes any 'input' positive

  • This is sometimes called the absolute value (of the input)

  • The modulus function is indicated by a pair of vertical lines being written around the input

    • Similar to how brackets are used
    • e.g. $\vert 7 \vert = 7 , \vert - 7 \vert = 7$

What is the relationship between a function and its modulus?

• For an 'output' such that $\mathrm{f} ( x ) \ge 0$, then $\vert \mathrm{f} ( x ) \vert = \mathrm{f} ( x )$

  • Both the function and its modulus are positive

• For an 'output' such that $\mathrm{f} ( x ) < 0$, then $\vert \mathrm{f} ( x ) \vert = - \mathrm{f} ( x )$

  • The function value is negative, but its modulus is positive