Relations, mappings and function notationIB MYP Maths Standard: Revision notes

A relation links inputs to outputs. We can show it as a mapping (arrows from inputs to outputs), a table, a list of ordered pairs such as $(1,4)$, or a graph. Example: the mapping $1\to4$, $2\to5$, $3\to5$ is the same as the table with inputs $1,2,3$ and outputs $4,5,5$, and the ordered pairs $(1,4)$, $(2,5)$, $(3,5)$.

A function is a relation in which every input has exactly one output.

• The mapping $1\to4$, $2\to5$, $3\to5$ is a function. Two different inputs can share an output.
• The mapping $1\to4$, $3\to5$, $3\to6$ is not a function, because input $3$ has two outputs. On a graph, use the vertical line test: if any vertical line crosses the graph more than once, it is not a function.

We write a function as $f(x)=3x-4$. This is read '$f$ of $x$ equals $3x-4$'. The letter $f$ is the name of the function and $x$ is the input. $f(x)$ is the output. Other letters can be used, such as $g(x)$, $h(t)$ or $C(d)$. In $C(d)$ the name $C$ may stand for cost and $d$ for distance.

To find $f(a)$, substitute $a$ in place of $x$ everywhere. For $f(x)=3x-4$: $f(5)=3(5)-4=11$, and $f(-2)=3(-2)-4=-10$. You can also substitute an expression: $f(2a)=3(2a)-4=6a-4$. For $g(x)=x^2+1$: $g(3)=3^2+1=10$. Use brackets for negatives: $g(-3)=(-3)^2+1=10$.

$f(a)$ means 'put $a$ in'. $f(x)=c$ means 'the output is $c$, find the input'. Write the equation and solve it. Example: $f(x)=3x-4$ and $f(x)=8$. Then $3x-4=8$, so $3x=12$ and $x=4$. Check: $f(4)=12-4=8$.

Function notation can describe real situations. If a taxi charges $C(d)=5+2d$ QAR for $d$ km, then $C(12)=29$ means a 12 km journey costs 29 QAR. $C(0)=5$ is the fixed starting charge, and $2$ is the cost per kilometre. To solve $C(d)=33$: $5+2d=33$, so $d=14$ km. Always say what your answer means, with units. Be careful: a model may stop being realistic for inputs far outside the values it was designed for.