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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.

Tutorial

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.

A function function * f(x)* is said to have an inverse if there exists another function

Suppose that * f(x) = x^{2} + 1*, does this function an inverse? It turns out that there is an easy way to tell. If we can find two values of

When a function is such that no two different values of * x* give the same value of

A horizontal line intersects the graph of * f(x )= x^{2} + 1* at two points, which means that the function is

Now we must be a bit more specific. When we say that * f(x) = x^{2} + 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

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

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

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

"But Wait!" you might be saying, "Isn't the inverse of * x^{2}* the square root of

When * x>0* and

Now we see further examples.

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