Composite and Inverse Functions
Compose functions in the stated order, track domain and range and reverse a one-to-one rule to obtain its inverse.
How to study Higher Mathematics
Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.
Core concepts
Concept 1
function notation
Exam cue: Which function acts first in the composition?
Concept 2
domain and range
Exam cue: What values are allowed before and after the mapping?
Concept 3
composition
Exam cue: Does the proposed inverse undo the original rule?
Concept 4
inverse functions
Concept 5
inverse graphs
Risk pitfalls and guardrails
Reading f(g(x)) in the wrong order
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Ignoring domain restrictions
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Confusing a reciprocal with an inverse function
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Memory anchors
Inside acts first
In f(g(x)), evaluate g before applying f.
Inverse undoes
Check f⁻¹(f(x)) = x on the valid domain.
Swap, then solve
Set y = f(x), interchange x and y, then isolate y.
Domain travels
Composition requires outputs of the inner function to fit the outer domain.
Inverse swaps sets
The original range becomes the inverse domain.
Reflection line
Inverse graphs reflect in y = x when the inverse exists.
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
Let f(x) = 2x + 3 and g(x) = x². Find f(g(2)).
Let f(x) = 3x − 1 and g(x) = x + 4. Find g(f(−2)).
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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