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THE VISUAL LEARNING STUDIO
See the idea. Understand the connection.
35 visual explanations across 29 chapters, with topic maps throughout the library.
The chapter index will grow as additional notes are prepared. Chapters shown here are a learning collection, not a claim that the entire exam syllabus is complete.
SEE IT. CONNECT IT. EXPLAIN IT.
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Study the expanded notes and the credited detailed textbook lessons. The current chapter index covers core topics; it is not a claim of exhaustive coverage of every examination syllabus. Check the official syllabus and primary references.
Functions — lesson overview
Definition and domain
A function assigns exactly one output to each input in its domain. Different inputs may share an output, but one input cannot have two different assigned outputs.
The domain is the allowed input set, the codomain is a specified target set and the range consists of outputs actually attained.
For f(x) = 1/x over real numbers, zero is excluded from the domain. A formula must be interpreted with its domain, not treated as unrestricted.
Types and composition
An injective function gives distinct outputs to distinct inputs. A surjective function reaches every member of its codomain; a bijection has both properties.
Whether a function is onto depends on its codomain. The same formula can be onto one codomain and not onto another.
Composition applies functions in sequence: (f∘g)(x) = f(g(x)), provided the relevant outputs of g lie in f’s domain.
Inverse and example
A two-sided inverse exists for a bijection between the specified sets. Restricting a domain can sometimes make an otherwise non-injective formula invertible.
For f(x)=2x+3 on the reals, solving y=2x+3 gives f⁻¹(y)=(y−3)/2.
The notation f⁻¹ is an inverse function here, not automatically the reciprocal 1/f(x).
