Types of Functions

Functions can be classified in two independent ways: by how inputs map to outputs (one-one, many-one, onto, into), and by the algebraic form of the rule (constant, linear, quadratic, polynomial). A Gujarat Std-11 Statistics student needs to recognise both.

Classification by mapping behaviour

Let $f : A \to B$.

• One-one (injective) function: different elements of $A$ always have different images in $B$. No two inputs share an output.
• Many-one function: at least two different elements of $A$ have the same image in $B$.
• Onto (surjective) function: every element of $B$ is the image of at least one element of $A$ — i.e. range $=$ co-domain.
• Into function: at least one element of $B$ is not the image of any element of $A$ — i.e. the range is a proper subset of the co-domain.
• Constant function: every element of $A$ maps to the same single element of $B$, e.g. $f(x) = 5$ for every $x$.
• Identity function: maps every element to itself, $f(x) = x$.

A function is always exactly one of {one-one, many-one} and exactly one of {onto, into} — the two classifications are independent, so a function can be, for instance, "one-one and into" or "many-one and onto".

Classification by algebraic form