Language of Functions (Cambridge (CIE) O Level Additional Maths): Revision Note

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 $\mathrm{f}\left(x\right),\mathrm{g}\left(x\right)$, etc

  • e.g.  $\mathrm{f}\left(x\right)=x^2-3x+2$ 

  • These would be pronounced as 'f of x', 'g of x', etc

• There is an alternative notation

  • e.g.  $\mathrm{f}:x\mapsto x^2-3x+2$

  • 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 $\left(x\right)$ and output $\left(\mathrm{f}\left(x\right)\mathrm{or}y\right)$

• 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 $\mathrm{f}\left(x\right)=2x+1$

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

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

    • $\mathrm{f}\left(a\right)=2a+1$

• 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 $\mathrm{f}\left(x\right)=15$, the equation $2x+1=15$ can be formed

    • Solving this equation gives an input of 7

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 $\mathrm{f}\left(x\right)=\sqrt{x}$ would be $x\ge 0$

• 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

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

• Graphing the function can help you visualise the range

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

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.  $\left|7\right|=7,|-7|=7$

What is the relationship between a function and its modulus?

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

  • Both the function and its modulus are positive

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

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