Functions, Graphs and Transformations
Working with domain, range, composite and inverse functions, modulus and reciprocal graphs, transformations and graphical equation solving.
How to study A-level Mathematics
Define the objects and conditions, select a representation, execute a justified method, then validate and interpret the result.
Core concepts
Concept 1
A function maps each input in its domain to exactly one output, with range determined by the rule and domain.
Exam cue: State the domain and range whenever an inverse or restriction is material.
Concept 2
An inverse reverses a one-to-one function on its stated domain, while composition applies functions in the written order.
Exam cue: Work from the inside outward for a composite function.
Concept 3
Graph transformations change inputs or outputs and must be distinguished from algebraic changes to the function itself.
Exam cue: Track a reference point and asymptotes when transforming a graph.
Risk pitfalls and guardrails
Assuming every function has an inverse over its full domain.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Reversing the order of a composition.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Applying a horizontal scale factor in the same direction as the coefficient inside the function.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Memory anchors
Domain
The domain is the permitted set of inputs to a function.
Range
The range is the set of outputs produced from the stated domain.
Composite Function
The composite fg applies g first and then f.
Inverse Function
An inverse function reverses a one-to-one mapping.
Asymptote
An asymptote is a line approached by a graph according to its limiting behaviour.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
For f(x)=3x−5, find f(4).
Let f(x)=x²+1 and g(x)=2x−3. Find fg(2), where fg means f∘g.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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