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Bijective Functions and Function Inverses

Bijective Functions and Function Inverses

Author: c o
Description:

To define the concept of an injective function
To define the concept of a surjective function
To define the concept of a bijective function
To define the inverse of a function

In this packet, the learning is introduced to the terms injective, surjective, bijective, and inverse as they pertain to functions.

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Tutorial

Domain, Range, and Back Again: A Function's Tale

Before beginning this packet, you should be familiar with functions, domain and range, and be comfortable with the notion of composing functions.  One of the examples also makes mention of vector spaces.

Introduction

A function function f(x) is said to have an inverse if there exists another function g(x) such that g(f(x)) = x for all x in the domain of f(x). In this case, g(x) is called the inverse of f(x), and is often written as f-1(x).  We will think a bit about when such an inverse function exists.

Suppose that f(x) = x2 + 1, does this function an inverse?  It turns out that there is an easy way to tell.  If we can find two values of x that give the same value of f(x), then the function does not have an inverse.  For instance, x = -1 and x = 1 both give the same value, 2, for our example.  Hence, f(x) does not have an inverse.

Injective Functions

When a function is such that no two different values of x give the same value of f(x), then the function is said to be injective, or one-to-one.  Injective functions can be recognized graphically using the 'horizontal line test':

A horizontal line intersects the graph of f(x )= x2 + 1 at two points, which means that the function is not injective (a.k.a. it is not one-to-one).  When no horizontal line intersects the graph at more than one place, then the function usually has an inverse.

Surjective Functions

Now we must be a bit more specific.  When we say that f(x) = x2 + 1 is a function, what do we mean?  We mean that it is a mapping from the set of real numbers to itself, that is f maps R to R.  But does f map all of R to all of R, that is, are there any numbers in the range that cannot be mapped by f?  Here is what I mean.  Are there any real numbers x such that f(x) = -2, for example?  The answer is no, there are not -  no matter what value we plug in for x, the value of f(x) is always positive, so we can never get -2.  

When a function maps all of its domain to all of its range, then the function is said to be surjective, or sometimes, it is called an onto function.  The best way to test for surjectivity is to do what we have already done - look for a number that cannot be mapped to by our function.  

Bijections

An example of a surjective function would by f(x) = 2x + 1; this line stretches out infinitely in both the positive and negative direction, and so it is a surjective function.  In a sense, it "covers" all real numbers.  Likewise, this function is also injective, because no horizontal line will intersect the graph of a line in more than one place.  Here is a picture

When a function, such as the line above, is both injective and surjective (when it is one-to-one and onto) it is said to be bijective.  A bijective function sets up a perfect correspondence between two sets, the domain and the range of the function - for every element in the domain there is one and only one in the range, and vice versa.  

It is clear then that any bijective function has an inverse.  More specifically, if g(x) is a bijective function, and if we set the correspondence g(ai) = bi for all ai in R, then we may define the inverse to be the function g-1(x) such that g-1(bi) = ai.  Such a function exists because no two elements in the domain map to the same element in the range (so g-1(x) is indeed a function) and for every element in the range there is an element in the domain that maps to it.

But Wait!

"But Wait!" you might be saying, "Isn't the inverse of x2 the square root of x?  According to what you've just said, x2 doesn't have an inverse."  The answer is "yes and no."   If a function doesn't have an inverse on its whole domain, it often will on some restriction of the domain.  For instance, if we restrict the domain to x > 0, and we restrict the range to y>0, then the function suddenly becomes bijective.  here is a picture:

When x>0 and y>0, the function y = f(x) = x2 is bijective, in which case it has an inverse, namely, f-1(x) = x1/2

 

Now we see further examples.

Examples

In this video we see three examples in which we classify a function as injective, surjective or bijective.