Functions

Properties of Functions
Terms	Description (with some alternate ways of looking at the definition)	Examples
domain, pre-image	An element from the first set.	x, f^-1(y)
range, image, codomain	An element in the second set.	y, f(x)
One-to-one, injection	For every value in the first set, there is a different value in the second set. If you can pass a horizontal line through the function and not hit more than one point. If no y has two or more x's. The y's are unique. If the "inverse" is not unique, it is NOT one-to-one and NOT an inverse. For finite sets, if cardinality of \|Domain\|>\|Range\|, then the function can't be one-to-one since some some member of the range has to have more than one mapping from the domain	YES	NO
		For integers, f(n)=n-1 Given any n, there is only one value of n+1.	For integers, f(n)=n²+1 both f(-1) and f(1)=2 Therefore 2 is not unique. NOTE: If we restricted the domain to positive values of n, then it would be one-to-one.
onto, surjection	For every value in the second set, there is a value in the first set. If every value in the range has a value in the domain. If the "inverse" does not map every value of the range to the domain, then it is NOT onto and NOT an inverse. For finite sets, if cardinality of \|Domain\|<\|Range\|, then the function can't be onto since some some member of the range will not be mapped into the domain since there are more elements in the range than the domain.	For integers, f(n)=floor(n/2) all values of floor(n/2) return values value in the set of integers.	For integers, f(n)=n²+1 Since n²+1 is always positive the range excludes part of the set of integers. NOTE if we restricted the range to positive integers greater than 1, it would be onto.
bijection, one-to-one correspondence, invertible	A function is both one-to-one AND onto. There is a unique value in the range for every value in the domain (on-to-one). Every value in the range had a value in the domain (onto). Going from the range to the domain is a function AND going from the domain to the range is a function. one-to-one tells us that there is a unique inverse, onto tells us that every element in the range has a inverse. For finite sets, if cardinality of \|Domain\|=\|Range\|, the function is onto if and only if it is one-to-one. Each elements in the domain is mapped into a unique element in the range and each element in the range is mapped into an element in the domain. NOTE: \|Domain\|=\|Range does not always imply this.	f(x)=x+1