Function Notation and Types (AA SL)
Before you can transform or sketch a graph, you need to be fluent in reading one: what \(f(x)\) means, and how to pull intercepts, turning points and asymptotes straight off an equation without a diagram to guide you. This page drills that skill across the function types AA SL uses most - quadratics, cubics and rationals - as a foundation for the broader Graphs & Transformations topic.
13 questions on this sub-topic.
Reading a function
Covered under IB syllabus references SL2.2 and SL2.4. Neither of these is a formula-booklet entry - they're definitions and a checklist of features you're expected to extract from an equation on sight.
Function notation
\(f(x)\)
\(f(x)\) is the output of \(f\) for input \(x\). \(f(0)\) gives the \(y\)-intercept; solving \(f(x)=0\) gives the \(x\)-intercept(s). The domain is every valid input; the range is every value the output can take.
Key features to find
Intercepts, turning points, asymptotes
\(x\)-intercepts (\(f(x)=0\)), \(y\)-intercept (\(f(0)\)), turning points (from a completed square or \(f'(x)=0\)), and any vertical or horizontal asymptotes.
Need graph transformations or the syllabus wording for this section? See Graphs & Transformations.
Worked examples
Let \(f(x)=x^2-5x+6\).
(a)(i) Find the \(x\)-intercept with \(x<2.5\).
(a)(ii) Find the \(x\)-intercept with \(x>2.5\).
(b) Find the \(y\)-intercept.
(c) Write down the equation of the axis of symmetry.
Worked solution
(a) \(x^2-5x+6=0\Rightarrow(x-2)(x-3)=0\Rightarrow x=2\) or \(x=3.\) M1
\(x=2.\) A1
A1
(b) \(f(0)=6.\) The \(y\)-intercept is \((0,6).\) A1
(c) \(x=\dfrac{2+3}{2}=2.5.\) A1
Let \(f(x)=x^3-3x^2-x+3\).
(a)(i) Find the \(x\)-intercept with \(x<0\).
(a)(ii) Find the \(x\)-intercept with \(0<x<2\).
(a)(iii) Find the \(x\)-intercept with \(x>2\).
(b)(i) Write down the coordinates of the local maximum.
(b)(ii) Write down the coordinates of the local minimum.
Worked solution
(a) Using GDC (or factorisation): \(f(x)=(x-3)(x-1)(x+1)\), so \(x=-1,\,1,\,3.\) M1 A1
(b) Solving \(f'(x)=3x^2-6x-1=0\) gives \(x=\dfrac{6\pm\sqrt{48}}{6}=1\pm\dfrac{2\sqrt3}{3}\), i.e. \(x\approx-0.155\) or \(x\approx2.15\). Local maximum \(\approx(-0.155,\,3.08)\), local minimum \(\approx(2.15,\,-3.08).\) A1 A1
Common mistakes
- Mixing up \(f(0)\) and \(f(x)=0\). \(f(0)\) is a single number - the \(y\)-intercept. Solving \(f(x)=0\) is a whole equation that can have several solutions - the \(x\)-intercept(s). Students regularly solve the wrong one under time pressure.
- Misreading the vertex from completed-square form. If \(f(x)=(x-h)^2+k\), the turning point is \((h,k)\), not \((-h,k)\) - the sign inside the bracket flips when you read off the coordinate.
- Assuming every function looks the same. A quadratic has at most one turning point and two roots; a cubic can have two turning points and up to three roots. Don't force a cubic's key features into a parabola-shaped mental template.
Ready to practise properly?
14 function-notation questions, marked instantly like the real exam.
Quick answers
What does f(x) actually mean?
\(f(x)\) is the output (\(y\)-value) that the function \(f\) produces for the input \(x\). Writing \(f(3)=7\) means the point \((3,7)\) lies on the graph of \(y=f(x)\).
What is the difference between the x-intercept and the y-intercept?
The \(y\)-intercept is found by evaluating \(f(0)\). The \(x\)-intercept(s) are found by solving \(f(x)=0\). They use the same equation in opposite directions, so mixing them up is one of the most common exam errors.