Functions

Also see: Polynomials

Definitions

Injective - A function is injective if it is one-to-one - that is, every element of the domain maps to at most one element of the range. (Note that an injective function may leave items in the range unmapped-to.)

Bijective - A function is bijective if it is BOTH one-to-one AND onto. That is, every element of the range is mapped to by exactly one element in the domain. A bijective function is both injective and surjective.

Surjective - A function is surjective, or onto, if every element of the range is mapped to by at least one element of the domain.

Algebra/Precalculus: Topics in Functions

Intermediate algebra functions outline:

• Introduction to functions
• Function notation
• Algebra and function composition
• Slope and rate of change
• Linear functions
• (Later chapters cover other kinds of functions)

Precalculus outline:

• sets and maps
• function definitions
• function notation
• domain and range
• rate of change
• composition
• transformation
• absolute value
• inverse functions
• conic sections - not really function stuff

More About Functions

Like Polynomials, functions start simple enough, but quickly get complicated, and can absorb hours, days, or even lifetimes of research. Functions are a way of mapping inputs to outputs, and they are immensely useful in using mathematics to describe, model, and understand the world.

Most college-level mathematics courses begin with functions, or arrive at the topic in short order. Functions provide the entire framework for precalculus, and when students hit calculus, it's all about continuous functions - the three conditions of continuity (calculus pop quiz: what are the three conditions of a function $f(x)$ being continuous at a location $x=a$?) being one of the first topics covered in calculus and one of the principal reasons calculus teachers spend so much time on limits.

(Answer: the three conditions for a function f(x) to be continuous at x = a are:

• $f(a)$ must exist
• $\lim _{x\rightarrow a+}f(x)=\lim _{x\rightarrow a-}f(x)=\lim _{x\rightarrow a}f(x)$
• $\lim _{x\rightarrow a}f(x)=f(a)$

These three criteria form the criteria for a continuous function. If a function is continuous, the limit of the function as x approaches a can be computed by simply evaluating the function at a.)

The study of functions continues in an undergraduate mathematics education with function analysis, split into real analysis and complex analysis. Engineering education focuses on the more practical aspects of functions - computing with software or calculators in the context of solving an applied problem.