Function Concepts (AI SL)
A function is a rule that turns every valid input into exactly one output, and function notation - like \(f(x)\) or \(C(n)\) - is the shorthand for describing that rule. This topic covers the domain and range of a function, what function notation means and how to evaluate it, and how functions are used as models for real situations, from taxi fares to fuel gauges.
What the syllabus says
This topic maps onto one point in the official IB Applications & Interpretation syllabus, within the Functions & Modelling unit.
| Code | Syllabus content |
|---|---|
| SL2.2 | Concept of a function, domain, range and graph. Function notation, for example \(f(x)\), \(v(t)\), \(C(n)\). The concept of a function as a mathematical model. The domain will be the largest possible domain for which a function is defined unless otherwise stated. Informal concept that an inverse function reverses or undoes the effect of a function; inverse function as a reflection in the line \(y=x\), with notation \(f^{-1}(x)\). |
This is core AI SL content, tested through both direct evaluation questions and modelling contexts.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is the domain of a function?
The domain is the set of valid input values a function can accept. For a real-world model, it's usually restricted to values that make sense in context, and the restriction is often given directly in the question.
e.g. For \(C(t)=25+18t\) with \(0\le t\le6\), the domain is \(0\le t\le6\).
What is the range of a function?
The range is the set of output values a function actually produces over its domain. For a straight-line model, you can find it by evaluating the function at both ends of the domain.
e.g. With \(C(t)=25+18t\) on \(0\le t\le6\): \(C(0)=25\), \(C(6)=133\), so the range is \(25\le C\le133\).
What does function notation like \(f(x)\) mean?
\(f(x)\) means "the output of function \(f\) when the input is \(x\)". To evaluate it at a specific value, substitute that value in place of \(x\) throughout the rule and simplify.
e.g. For \(F(d)=60-0.08d\): \(F(200)=60-0.08(200)=60-16=44.\)
How do you evaluate a function at a given input?
Replace the variable with the given value everywhere it appears in the rule, then work through the arithmetic in the usual order of operations to get a single output value.
e.g. For \(f(x)=2x+3\): \(f(6)=2(6)+3=15.\)
What does it mean for a function to model a real context?
A function models a context when its input, output, and any domain restriction all correspond to real quantities - the value \(f(0)\) often has a specific practical meaning, such as a fixed starting charge.
e.g. In a call-out-fee model \(C(t)=25+18t\), \(C(0)=\$25\) is the fee charged even before any hours are worked.
Key formulas
This topic is built more on definitions than formulas - the table below sets out the key notation, and the comparison and cards underneath unpack each idea.
Formula reference
None of these are formulas to memorise from the booklet - they're the notation and definitions this whole topic runs on.
| Notation | Used for | Booklet? |
|---|---|---|
| \(f(x)\), \(v(t)\), \(C(n)\) | Function notation - naming the rule and its variable | Not in booklet - notation, not a formula |
| Domain: the set of valid \(x\) | Where the function is defined | Not in booklet - definition, prior knowledge |
| Range: the set of resulting \(f(x)\) | What values the function outputs | Not in booklet - definition, prior knowledge |
Domain vs range
Domain and range describe the same function from opposite ends - one is about what goes in, the other about what comes out.
| Feature | Domain | Range |
|---|---|---|
| What it describes | Valid input (\(x\)) values | Possible output (\(f(x)\)) values |
| How to find it (restricted linear model) | Read the stated restriction on \(x\) | Substitute the domain endpoints into \(f\) |
| Example (window cleaner, \(0\le t\le6\)) | \(0\le t\le6\) | \(25\le C\le133\) |
Domain and range
The two ideas that define exactly which inputs and outputs a function covers.
Domain
The largest set of \(x\)-values for which the function is defined, unless the question restricts it further.
Range
The set of output values the function actually reaches - for a linear model, check both endpoints of the domain.
Restricted (real-world) domains
A model's domain is usually restricted to values that make physical sense - time \(\ge0\), a distance that can't exceed a fixed maximum, and so on.
Function notation and evaluation
Reading and using \(f(x)\) correctly is the single most-tested skill in this topic.
What \(f(x)\) means
\(f(x)\) is the output of the rule \(f\) for input \(x\) - it names the relationship, not a multiplication of \(f\) and \(x\).
Evaluating at a point
Substitute the given value for the variable everywhere it appears, then simplify - the result is a single number.
Reading values from a graph or table
A graphed or tabulated function lets you read off \(f(x)\) for a given \(x\), or the reverse: which \(x\) gives a stated output.
Functions as models
Once a function represents something real, every part of it - inputs, outputs, and domain - carries meaning.
Interpreting \(f(0)\)
In many cost or charge models, \(f(0)\) is the fixed or starting value before any variable cost is added.
Checking the model's limits
A linear model can produce nonsensical values outside its stated domain - always check whether an answer falls inside the range the context actually allows.
Using your GDC to visualise
Graphing the model makes the domain, range, and shape of the relationship immediate, rather than something you have to compute by hand.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
A window cleaner charges a call-out fee plus an hourly rate. The cost is modelled by \(C(t) = 25 + 18t\) dollars, where \(t\) is the number of hours worked, for \(0 \le t \le 6\).
(a) State the domain of \(C\).
(b) Find \(C(3)\), the cost of a 3-hour job.
(c) State the range of \(C\).
(d) Interpret the meaning of \(C(0)\) in this context.
Worked solution
(a) Domain: \(0 \le t \le 6.\) A1
(b) \(C(3) = 25 + 18(3) = 25 + 54\) M1
\(= $79.\) A1
(c) \(C(0)=25,\ C(6)=133,\) so range: \(25 \le C \le 133.\) A1
(d) \(C(0)=$25\) represents the call-out fee charged even if no hours are worked. A1
A taxi fare (in dollars) is modelled as a linear function \(f(x)\) of the distance travelled \(x\) km.
(a) Find the equation of the function \(f(x)\), given \(f(0)=3\) and \(f(5)=13.\)
(b) State the domain and range of \(f,\) given \(0 \le x \le 20.\)
(c) Find \(f(6).\)
(d) Determine whether the graph of \(f\) is parallel to the line \(y=2x-5.\)
Worked solution
(a) Gradient \(m=\dfrac{13-3}{5-0}\) M1
\(=2.\) A1
\(f(x)=2x+3.\) A1
(b) Domain: \(0 \le x \le 20.\) A1
(c) \(f(6)=2(6)+3=15.\) M1
Range: \(3 \le f(x) \le 43.\) A1
(d) The line \(y\) R1
\(=2x-5\) also has gradient \(2,\) so the graphs are parallel. A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Confusing \(f(x)\) with \(f\times x\). \(f(x)\) is function notation for "the output of \(f\) at \(x\)" - it isn't multiplication, and \(f(3)\) means substitute \(x=3\), not "\(f\) times 3".
- Giving the range using the domain's numbers. The range is the set of \(y\)-values, not \(x\)-values - for a linear model, substitute the domain endpoints into the function to get the actual range.
- Ignoring a stated domain restriction. A model like \(F(d)=60-0.08d\) for \(0\le d\le700\) isn't valid outside that interval, even if the algebra still "works" - always check an answer stays within the given domain.
- Skipping the context when asked to interpret a value. "Interpret \(C(0)\)" wants a sentence about what that number means in the situation (e.g. a fixed fee), not just the numerical value restated.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
The starting point for almost every graphing task - if you can't get the graph on screen, nothing else works.
- Use \(x\) as the variable - type it with the dedicated \(x\) key, not ALPHA + X.
- Press Y= and type the function next to Y1=. Use X,T,θ,n for x. Press GRAPH to draw it. Clear old functions by moving to them and pressing CLEAR.TI-84
- Open a Graphs page (press ctrl + I → Add Graphs, or press the Graphs app). Type the function in the entry bar at the bottom and press ENTER.Nspire
- Press MENU → Graph (or press the Graph icon). Press SHIFT → F3 (TYPE) to choose the graph type (Y= is the default). Type the function next to Y1 and press F6 (DRAW).Casio
- If the graph looks blank or wrong: check the window and check the angle mode (degrees vs radians).
- To graph multiple functions, enter them as Y1, Y2, Y3 etc. - useful for finding intersections.
Tip: The most common reason a graph doesn't appear is a bad viewing window, not a mistake in the function. Try ZoomFit or ZStandard first.
Read off the exact \(y\)-value for a given \(x\) - useful for checking answers and for questions that ask for a specific coordinate.
- After graphing the function, you can evaluate it at any \(x\)-value.
- Method 1 - Table: 2nd → GRAPH (TABLE), scroll to the \(x\) you want. Method 2 - Trace: press TRACE, then type the \(x\)-value and press ENTER. Method 3 - Home screen: type Y1(value) e.g. Y1(3) using VARS → Y-VARS → 1:Function.TI-84
- On the graph, press TRACE and type the \(x\)-value, then ENTER. Or on a Calculator page type f1(3) to evaluate the stored function at \(x=3\).Nspire
- Press TRACE (F1) on the graph, then type the \(x\)-value and EXE. Or use the Table view (MENU → Table) to see multiple values.Casio
- For models: substitute the \(x\)-value into the regression equation stored in the calculator, if one was fitted.
Tip: Trace gives an approximate value by cursor position - typing the \(x\)-value after pressing TRACE gives the exact value.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Function concepts questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What is the difference between the domain and the range of a function?
The domain is the set of valid input (\(x\)) values; the range is the set of resulting output (\(f(x)\)) values. For a real-world model, both are usually restricted to the values that make sense in context.
What does \(f(3)\) mean?
\(f(3)\) means "substitute \(x=3\) into the function \(f\) and evaluate the result" - it gives a single output value, not an equation to solve.
How do I state the domain of a modelling question?
Look at what the input variable represents physically - time can't be negative, a percentage can't exceed 100, and so on. The question will often give you the restriction directly, such as "for \(0\le t\le6\)".
Can I use my GDC to explore a function?
Yes - graphing the function lets you see its shape, read off the range visually, and evaluate it at any point using a table or the trace feature, which is much faster than substituting by hand for several values.
Sub-topics
Function Concepts broken down into its individual skills, each with its own focused page.
Related topics
More Functions & Modelling topics from the same AI SL syllabus unit, in case you want to keep going.