3.2 Function notation
- Syllabus
- 2017
- Topic
- 3.2
- Level
- Higher
Learning objectives
3.2HA Functions as mappings between setsUnderstand a function as a mapping between elements of two sets.3.2HB Function notationuse function notation in the forms f(x) = … and f : x ↦ …3.2HC Domain and rangeunderstand domain and range, including values excluded from a domain, such as x = 2 for f(x) = 1/(x − 2)3.2HD Composite and inverse functionsunderstand and find the composite function fg and the inverse function f⁻¹; fg means apply g first, then f
Understand a function as a mapping
A function maps each allowed input to exactly one output. Different inputs may share an output, but one input cannot be assigned two different outputs.
| Idea | Meaning |
|---|---|
| input | an allowed element of the domain |
| output | the value assigned by the function |
| mapping | the rule connecting each input to its output |
| image | the output produced by a particular input |
A mapping diagram represents a function if every input has exactly one outgoing arrow. An output may receive several arrows, and some listed outputs may receive none.
The rule x↦x2 maps both −3 and 3 to 9; this is still a function because each input has only one output.
‘Exactly one output per input’ does not mean ‘exactly one input per output’. The latter extra condition is needed for an inverse to be a function on the full range.
Read, evaluate and write function notation
The notation f(x) names the output of function f when the input is x. The mapping form f:x↦3x−2 and the equation f(x)=3x−2 describe the same rule.
| Request | Action |
|---|---|
| find f(5) | substitute x=5 into the rule |
| find f(a+1) | replace every x by (a+1) |
| solve f(x)=k | set the rule equal to k and solve |
| write mapping form | state f:x↦ followed by the rule |
If f(x)=3x−2, then f(5)=3(5)−2=13 and f(a+1)=3(a+1)−2=3a+1.
Functions named f and g can use different rules. Keep the function name attached to the correct definition throughout a multi-part problem.
f(x) is not f multiplied by x. It is a single notation for the output produced by input x.
Control the domain and range of a function
The domain is the set of allowed inputs; the range is the set of outputs actually produced. Algebraic restrictions and stated intervals can limit either set.
| Feature | Domain control |
|---|---|
| denominator | exclude values making it zero |
| even root | require the radicand to be non-negative |
| stated interval | keep only inputs inside it |
| inverse function | its domain is the original range |
For f(x)=1/(x−2), x=2 makes the denominator zero, so the domain excludes 2. Also f(x) can never equal 0, so the range excludes 0.
On a graph, project the curve onto the x-axis for the domain and onto the y-axis for the range, respecting open endpoints, holes and asymptotes.
A value excluded from the domain is an input restriction; it is not automatically excluded from the range. Analyse inputs and outputs separately.
Compose functions and find inverses
A composite applies functions in sequence: fg(x)=f(g(x)), so g acts first. An inverse function reverses the original mapping and satisfies f−1(f(x))=x on the permitted domain.
| Task | Method |
|---|---|
| find fg(x) | substitute the whole expression g(x) into f |
| evaluate fg(a) | find g(a), then apply f |
| find f−1 | write y=f(x), solve for x, then swap labels |
| verify inverse | simplify both f−1(f(x)) and f(f−1(x)) |
If f(x)=2x+1 and g(x)=x2, then fg(x)=2x2+1, while gf(x)=(2x+1)2. Order matters.
For f(x)=3x−5, set y=3x−5 and solve x=(y+5)/3, so f−1(x)=(x+5)/3.
A many-to-one rule such as x2 needs a restricted domain, for example x≥0, before its inverse x is a function.
The superscript −1 denotes an inverse, not a reciprocal: generally f−1(x)=1/f(x).