Lecture 6 – Types of Functions
Inverse Functions
A function that does the opposite of the function from which it was formed. For a function f(x), its
inverse is written as f-1(x). For f-1(x), the domain of f(x) would be its codomain and the codomain of
f(x) would be its domain.
Some functions may not have an inverse function, unless their domain is restricted. One-to-many
functions cannot have inverse functions as they would become many-to-one functions, which aren’t
functions – but are still relationships.
For a function to have an inverse, they must be bijective (discussed below).
Surjective Functions
A function may be surjective if, for every element in A (a), there is at least one element in B (b)
where f(b) = a. In other words, the range of the function must equal the codomain, every image has
a preimage. To prove this:
For any number y there must be a number x such that f(x) = y. For example, if f(x) = x + 1, then y = x +
1 which rearranges to x = y – 1. So, because y is an integer, x must also be an integer, and so for any
integer, y, there is an integer x - 1 and so f(x) = y. Q.E.D.
Injective Functions
A function may be injective if, for every element in A (a), there is only one element in B (b) where
f(b) = a. This means that all images have one preimage and/or that there exist only one-to-one
relations. To prove this:
For any number x there must be a number y such that f(x) = f(y) and can be simplified to x = y. For
example, if f(x) = 2x the f(y) = 2y. If f(x) = f(y), then 2x = 2y, which can become x = y. Q.E.D.
Bijective Functions
Bijective functions are functions that are both surjective and injective. Bijective functions are able to
have inverse functions. A function is bijective if, f(a) = f(b) and we can show that a = b.
Inverse Functions
A function that does the opposite of the function from which it was formed. For a function f(x), its
inverse is written as f-1(x). For f-1(x), the domain of f(x) would be its codomain and the codomain of
f(x) would be its domain.
Some functions may not have an inverse function, unless their domain is restricted. One-to-many
functions cannot have inverse functions as they would become many-to-one functions, which aren’t
functions – but are still relationships.
For a function to have an inverse, they must be bijective (discussed below).
Surjective Functions
A function may be surjective if, for every element in A (a), there is at least one element in B (b)
where f(b) = a. In other words, the range of the function must equal the codomain, every image has
a preimage. To prove this:
For any number y there must be a number x such that f(x) = y. For example, if f(x) = x + 1, then y = x +
1 which rearranges to x = y – 1. So, because y is an integer, x must also be an integer, and so for any
integer, y, there is an integer x - 1 and so f(x) = y. Q.E.D.
Injective Functions
A function may be injective if, for every element in A (a), there is only one element in B (b) where
f(b) = a. This means that all images have one preimage and/or that there exist only one-to-one
relations. To prove this:
For any number x there must be a number y such that f(x) = f(y) and can be simplified to x = y. For
example, if f(x) = 2x the f(y) = 2y. If f(x) = f(y), then 2x = 2y, which can become x = y. Q.E.D.
Bijective Functions
Bijective functions are functions that are both surjective and injective. Bijective functions are able to
have inverse functions. A function is bijective if, f(a) = f(b) and we can show that a = b.
