Knowledge organisers

Functions: IB Maths AA SL knowledge organiser

Everything to know about functions on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Domain and range: The domain is the set of inputs a function accepts; the range is the set of outputs it gives.
Composite function: (f ∘ g)(x) = f(g(x)): apply g first, then f.
Inverse function: f⁻¹ undoes f. Its graph is the reflection of the graph of f in the line y = x.
Discriminant: For ax² + bx + c, Δ = b² − 4ac tells you how many real roots there are: two, one (repeated) or none.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Gradient; forms of a lineIn the formula booklet	\(m=\frac{y_2-y_1}{x_2-x_1},\) \(y=mx+c,\) \(ax+by+d=0,\) \(y-y_1=m(x-x_1)\)
Parallel; perpendicular lines	\(m_1=m_2;\) \(m_1m_2=-1\)
Quadratic: axis of symmetryIn the formula booklet	\(x=-\frac{b}{2a}\)
Quadratic formula; discriminantIn the formula booklet	\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},\) \(\Delta=b^2-4ac\)
Roots: \(\Delta>0\) two distinct real, \(\Delta=0\) one repeated, \(\Delta<0\) no real roots
Vertex form, vertex \((h,k)\); intercept form	\(y=a(x-h)^2+k;\) \(y=a(x-p)(x-q)\)
Composite; inverse	\((f\circ g)(x)=f\big(g(x)\big);\) \(f\big(f^{-1}(x)\big)=x\)
Graph of \(f^{-1}\): reflect \(y=f(x)\) in \(y=x\); domain of \(f^{-1}\) = range of \(f\)
Translations: \(f(x)+b\) up \(b\); \(f(x-a)\) right \(a\)
Stretches: \(pf(x)\) vertical ×\(p\); \(f(qx)\) horizontal ×\(\tfrac1q\)
Reflections: \(-f(x)\) in the \(x\)-axis; \(f(-x)\) in the \(y\)-axis
\(y=\frac{ax+b}{cx+d}\) asymptotes	\(x=-\frac dc,\) \(y=\frac ac\)
Exponential ↔ log	\(y=a^x\iff x=\log_ay;\) \(e^x\text{ and }\ln x\text{ are inverses}\)

Worked example

f(x) = 2x + 3 and g(x) = x². Find (g ∘ f)(1) and f⁻¹(x).

(g ∘ f)(1) = g(f(1)) = g(5) = 5² = 25
Let y = 2x + 3, swap x and y: x = 2y + 3, so y = (x − 3)/2

Answer: (g ∘ f)(1) = 25 and f⁻¹(x) = (x − 3)/2

Common mistakes

Answer in the wrong form: asymptotes not as equations, coordinates vs values, unfinished answers
Transformations and inverses: incomplete descriptions and f⁻¹ notation
Mixing up the perpendicular gradient rule (m₁·m₂ = −1)
Sign errors completing the square

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Find the gradient and midpoint of two points and write the equation of a line in gradient-intercept, general and point-gradient forms, such as y − 3 = 2(x − 1).
Sketch parabolas from their factorised, vertex and standard forms, marking intercepts, the vertex and the axis of symmetry.
Sketch y = f(x) + a and y = f(x + a) from a given graph, describing each as a translation by a vector and tracking key points.
Solve equations involving rational functions algebraically and with graphs, such as where (2x + 1)/(x − 3) meets a line, checking solutions against the domain.
Apply a sequence of transformations to a graph in the correct order and write the equation of the resulting graph.
Work through multi-part problems that combine transformations, inverse functions and differentiation, using results from earlier parts to answer later ones.

The printable sheet

Functions knowledge organiser for IB Maths AA SL: one A4 page of key definitions, formulas, a worked example and common mistakes — Functions knowledge organiser (IB Maths AA SL), A4. Download the PDF.

Revise it next

Other IB Maths AA SL topics: Number and algebra · Geometry and trigonometry · Statistics and probability · Calculus · All IB Maths AA SL organisers