Video for Homework H07.2 Injective, Surjective, Bijective, and Inverse Functions

Reading: Section 7.2 One-to-One Functions, Onto Functions, Inverse Functions

Homework: H07.2: 7.2#5,7,12,17,41,49

Topics: • Injective Functions • Surjective Functions • Bijective Functions • Inverse Functions

Injective Functions

Definition of Injective Function

Words: 푓 is injective, or 푓 is an injection, or 푓 is one-to-one Usage: 푓 is a function, 푓: 푋 → 푌 Meaning: If two inputs cause outputs that are equal, then the inputs must be equal. Meaning Written Formally: ∀푥1, 푥2 ∈ 푋(퐼푓 푓(푥1) = 푓(푥2) 푡ℎ푒푛 푥1 = 푥2) Contrapositive Meaning: If two inputs are not equal, then the outputs will not be equal. Contrapositive Written Formally: ∀푥1, 푥2 ∈ 푋(퐼푓 푥1 ≠ 푥2 푡ℎ푒푛 푓(푥1) ≠ 푓(푥2)) Other Wording: For every element in the codomain, there exists at most one element of the domain that can be used as input to cause that element of the codomain to be output.

Other Formal Presentation: ∀푦 ∈ 푌(∃ at most one 푥 ∈ 푋(푓(푥) = 푦))

Incorrect interpretation of the term one-to-one

When students are asked what one-to-one means, the answer they always give is always this:

For each 푥, there is one y. It is important to realize that the sentence above is not what it means to be one-to-one. In fact, the sentence above is what it means for 푓 to be a function. Every function has the property that for each 푥, there is one 푦. It has nothing to do with the property of being one-to-one.

Because the term one-to-one is so often misunderstood, many mathematicians prefer to use the term injective instead. In these videos, I will mostly use the term injective.

What it means to not be injective

It will be important to know how to determine when a function is injective or not injective. For that, it will be necessary to know the negation of the formal statement that 푓 is injective.

푓 is injective:

푓 is not injective:

[Example 1] f f f X Y X Y X Y

a p p a b q a q b p c r b r c q d s c s d r d t t e

f f f X Y X Y X Y

a p p a b q a q b p c r b r c q d s c s d r

c t t e

Graphs of functions and of injective and non-injective functions.

A function whose domain and codomain are subsets of the real numbers 푹 can be illustrated using a graph. But there are graphs that do not correspond to functions. The vertical line test articulates which graphs qualify to be graphs of functions

The Vertical Line Test for a Graph to be the Graph of a Function

• If, for every 푎 ∈ 퐴, the vertical line 푥 = 푎 intersects the graph exactly once, then the graph is the graph of a function with domain 퐴. (The graph passes the vertical line test.) • If there exists an 푎 ∈ 퐴 such that the vertical line 푥 = 푎 does not intersect the the graph, or intersects the graph more than once, then the graph is not the graph of a function with domain . (The graph fails the vertical line test.) 퐴

The injective or non-injective property of the function has corresponding behavior in the graph of the function. The correspondence is the essence of what is often called the horizontal line test. The name actually needs to be more precise.

The Horizontal Line Test for Injectivity Suppose 퐴 ⊆ 푹 and 퐵 ⊆ 푹 and 푓: 퐴 → 퐵 injectivity of 푓 ↔ behavior of the graph of 푓 For every 푏 ∈ 퐵,the horizontal line 푦 = 푏 intersects the 푓 is injective ↔ graph of 푓 at most once. (푓 passes the horizontal line test.) There exists a 푏 ∈ 퐵 such that the horizontal line 푦 = 푏 푓 is not injective ↔ intersects the graph of 푓 more than once. (푓 fails the test.)

Determining whether a function given by a formula is injective.

[Example 2] Let 푓: 푹 → 푹 be the function 푓(푥) = 푥2. (a) Is 푓 injective? (Is 푔 one-to-one?) Prove or give a counterexample. Illustrate using a graph.

[Example 3] Let 푓: 푹 → 푹 be the function 푓(푥) = 4푥 − 5. (a) Is 푓 injective? (Is 푓 one-to-one?) Prove or give a counterexample. Illustrate using a graph.

푥 − 2 [Example 4] Let 푓(푥) = 푥 − 3 (a) What is the domain of 푓(푥)? (the natural domain)

(b) What is the codomain of 푓(푥)?

(c) What is the range of 푓(푥)? That is, what is the following set? 푟푎푛푔푒(푓) = 푖푚푎푔푒 표푓 퐴 = 푓(퐴) = {푦 = 푓(푥)|푥 ∈ 퐴} To answer that question, it helps to think of the formula for 푓(푥) as an equation involving 푥 and 푦 and and to solve the equation for 푥.

(d) Is 푓 injective? (Is 푓 one-to-one?) Prove or give a counterexample.

(e) Illustrate your results from (a), (c), (d) using this given graph of 푓(푥).

Surjective Functions

Definition of Surjective Function

Words: 푓 is surjective, or 푓 is a surjection Alternate Words: 푓 is onto Usage: 푓 is a function, 푓: 푋 → 푌 Meaning: For every element in the codomain, there exists an element of the domain (at least one) that can be used as input to cause that element of the codomain to be output.

Meaning Written Formally: ∀푦 ∈ 푌(∃푥 ∈ 푋(푓(푥) = 푦))

What it means to not be surjective

It will be important to know how to determine when a function is surjective or not surjective. For that, it will be necessary to know the negation of the formal statement that 푓 is surjective.

푓 is surjective:

푓 is not surjective:

[Example 1](continued) Return to [Example 1] and indicate which functions are surjective.

Graphs of surjective and non-surjective functions. Earlier we saw that for a function whose domain and codomain are subsets of the real numbers, there is a horizontal line test for injectivity. Observe that it is possible to articulate a similar test for surjectivity.

The Horizontal Line Test for Surjectivity

Suppose 퐴 ⊆ 푹 and 퐵 ⊆ 푹 and 푓: 퐴 → 퐵 surjectivity of 푓 ↔ behavior of the graph of 푓 For every 푏 ∈ 퐵,the horizontal line 푦 = 푏 intersects the 푓 is surjective ↔ graph of 푓 at least once. (푓 passes the horizontal line test.) There exists a 푏 ∈ 퐵 such that the horizontal line 푦 = 푏 does 푓 is not surjective ↔ not intersect the graph of 푓. (푓 fails the test.)

Return to the functions presented in earlier examples and indicate whether each is surjective.

[Example 2](continued) Let 푓: 푹 → 푹 be the function 푓(푥) = 푥2. (b) Is 푓 surjective? (Is 푓 onto?) Prove or give a counterexample. Illustrate using a graph.

[Example 3](continued) Let 푓: 푹 → 푹 be the function 푓(푥) = 4푥 − 5. (b) Is 푓 surjective? (Is 푓 onto?) Prove or give a counterexample. Illustrate using a graph.

푥 − 2 [Example 4](continued) Let 푓(푥) = 푥 − 3 (f) Is 푓 surjective? (Is 푓 onto?) Prove or give a counterexample.

(g) Illustrate your result from (f) using this given graph of 푓(푥).

A function’s properties depend on the choice of domain and codomain.

[Example 4](c) Recall that the function from [Example 4] 푥 − 2 푓(푥) = 푥 − 3 has domain 퐴 = 푹 − {3} = {푥 ∈ 푹|푥 ≠ 3}

In function notation, we write 푓: 퐴 → 푹

Notice that we cannot write 푓: 푹 → 푹, because the formula for 푓(푥) does not give a 푦 value when 푥 = 3. The domain cannot be the set of all real numbers. We would say that the formula 푥 − 2 푓(푥) = 푥 − 3 does not give a well-defined function on the set of all real numbers. It fails to do what a function is required to do.

The conclusion from this is that the choice of domain is important when describing a function. Also recall that we found that 푓 was not surjective: For the number 푦 = 1 in the codomain, there is no 푥 such that 푓(푥) = 1.

We can describe this using the terminology of the range and the codomain.

• The codomain of 푓 is the set of all real numbers 푹. • The range of 푓 is the set 푹 − {1} = {푦 ∈ 푹|푦 ≠ 1} We see that the range is not equal to the codomain. (The range is a proper subset of the codomain.)

푟푎푛푔푒(푓) ⊊ 푐표푑표푚푎푖푛(푓)

Realize that if we define a set 퐵 as follows 퐵 = 푹 − {1} = {푦 ∈ 푹|푦 ≠ 1}

And then use just the set 퐵, rather than all of 푹, for the codomain, then the resulting function 푓: 퐴 → 퐵 is surjective!

The conclusion from this is that the choice of codomain is important when describing a function.

Now return to the function from [Example 2], 푓: 푹 → 푹 defined by 푓(푥) = 푥2. We found that the function was neither injective nor surjective.

But observe that if we define the set 퐴

and define the set 퐵

Then the function 푓: 퐴 → 퐵 defined by 푓(푥) = 푥2 is both injective and surjective!

[Example 3](continued)

(c) Now return to the function from [Example 3], 푓: 푹 → 푹 defined by 푓(푥) = 4푥 − 5. What if we change the domain and codomain to the set 풁. That is, we have

푓: 풁 → 풁 defined by 푓(푛) = 4푛 − 5

Then

Bijective Functions

Definition of Bijective Function

Words: 푓 is bijective, or 푓 is a bijection or 푓 is a one-to-one correspondence Usage: 푓 is a function, 푓: 푋 → 푌 Meaning: 푓 is both injective and surjective. (푓 is both one-to-one and onto.)

What it means to not be bijective

It will be important to know how to determine when a function is bijective or not bijective.

For that, it will be necessary to know the negation of the statement that 푓 is bijective.

푓 is bijective:

푓 is not bijective:

[Examples 1,2,3,4] continued Return to the functions presented in [Examples 1,2,3,4] and indicate whether each is bijective.

What would be a formal presentation of the meaning of bijective?

To answer that question, we should review the definitions of surjective and injective.

풇 is surjective: For every element in the codomain, there exists at least one element of the domain that can be used as input to cause that element of the codomain to be output.

풇 is injective: For every element in the codomain, there exists at most one element of the domain that can be used as input to cause that element of the codomain to be output.

We can combine these into one concise statement:

풇 is bijective: For every element in the codomain, there exists exactly one element of the domain that can be used as input to cause that element of the codomain to be output.

Meaning of Bijective Written Formally:

In light of this discussion, it is worthwhile to add some lines to our definition of bijective:

Definition of Bijective Function (updated with new lines)

Words: 푓 is bijective, or 푓 is a bijection or 푓 is a one-to-one correspondence Usage: 푓 is a function, 푓: 푋 → 푌 Meaning: 푓 is both injective and surjective. (푓 is both one-to-one and onto.) Other Wording: For every element in the codomain, there exists exactly one element of the domain that can be used as input to cause that element of the codomain to be output.

Meaning Written Formally: ∀푦 ∈ 푌(∃! 푥 ∈ 푋(푓(푥) = 푦))

Inverse Maps and their graphs

Definition of the Inverse Map

Given function 푓: 푋 → 푌, we define the inverse map 푓−1: 푌 → 푋 by saying that 푓−1(푦) = 푥 means 푓(푥) = 푦 In terms of the graph, this tells us that

the point (푏, 푎) is on the graph of 푓−1 whenever the point (푎, 푏) is on the graph of 푓

To make a graph of 푦 = 푓−1(푥) by hand • Start with a graph of 푦 = 푓(푥) and a second, blank 푥, 푦 axes for the graph of 푓−1 • Find a key point (푥, 푦) = (푎, 푏) on the graph of 푓(푥), and interchange the 푥, 푦 coordinates to get a new ordered pair (푏, 푎). Plot the ordered pair (푥, 푦) = (푏, 푎) on the axes for 푓−1. • Repeat this for a bunch of key points.

• When you have enough key points plotted, sketch the graph of −1. 푓 This is equivalent to doing the following:

• Start with a graph of 푦 = 푓(푥) and flip the graph across the line 푦 = 푥 to obtain the graph of −1 푓

Examples of graphs of some functions and their inverse maps

When is an inverse map qualified to be called a function, and what properties will the inverse function have?

Observe from the three examples above, that given a function 푓. • The graph of the 푓−1 will pass the vertical line test for function. (that is, 푓−1 will be a function) whenever graph of 푓 passes both horizontal line tests (that is, whenever 푓 is both onto and one-to-one).

• The graph of 푓−1 will automatically pass both horizontal line tests (that is, 푓−1 will automatically be both onto and one-to-one) because the graph of passes the vertical line 푓 test (that is, because 푓 is a function).

We have convinced ourself of the truth of Theorem 7.2.2 and Theorem 7.2.3

Theorem 7.2.2 When an Inverse Map will be a Function

If a function 푓: 푋 → 푌 is bijective, then the inverse map 푓−1: 푌 → 푋 defined by saying 푓−1(푦) = 푥 means 푓(푥) = 푦 will have the qualifications to be called a function.

Definition of the Inverse Function

If a function 푓: 푋 → 푌 is bijective, then the inverse map 푓−1: 푌 → 푋 (which is guaranteed to be a function by Theorem 7.2.2) is called the inverse function for 푓.

Theorem 7.2.2 The Inverse Function will be Both Injective and Surjective

If a function 푓: 푋 → 푌 is bijective, then its inverse function 푓−1: 푌 → 푋 will also be bijective. . The book gives a proof of Theorem 7.2.2, but I find that a much simpler proof is one of the coolest things ever. Here is my proof of Theorem 7.2.2 Proof

Suppose that a function 푓: 푋 → 푌 is bijective. Then we have two quantified statements that are true about 푓:

• Because 푓 is a function, we know that ∀푥 ∈ 푋 (∃! 푦 ∈ 푌(푦 = 푓(푥)))

• Because 푓 is bijective, we know ∀푦 ∈ 푌 (∃! 푥 ∈ 푋(푦 = 푓(푥)))

But by definition of inverse map 푓−1: 푌 → 푋, the expression 푦 = 푓(푥) means same thing as the expression 푥 = 푓−1(푦).

Replacing the expression 푦 = 푓(푥) with the expression 푥 = 푓−1(푦) in the above statements,

• We know ∀푥 ∈ 푋 (∃! 푦 ∈ 푌(푥 = 푓−1(푦))). This tells us that 푓−1: 푌 → 푋 is bijective.

• We know ∀푦 ∈ 푌 (∃! 푥 ∈ 푋(푥 = 푓−1(푦))). This tells us that 푓−1: 푌 → 푋 is a function. End of Proof

Finding a formula for the inverse function

Think of the formula for 푓(푥) as a machine that takes x as input and spits out y as output. That is,

푦 = 푓(푥)

This gives you an equation involving 푥 and 푦, an equation that is solved for 푦 in terms of 푥.

Solve the equation for 푥 in terms of 푦.

Think of the new equation as a machine that takes 푦 as input and spits out 푥 as output. That equation describes 푓−1 as a function of 푦. That is, 푥 = 푓−1(푦) The resulting expression for 푓−1(푦) can be rewritten using another variable, if you want.

Remark on changing the variable names

Students often tell me that they have learned to find the inverse function by this process

• interchange 푥 and 푦 in the equation • solve for 푦 Some students prefer this, because it produces a function 푓−1(푥) with variable 푥, not 푦.

I prefer my method, of having

• function 푦 = 푓(푥) • inverse function 푥 = 푓−1(푦)

This presentation makes it clear that • is a function that takes as input and poduces the output . 푓: 푋 → 푌 푥 푦 • −1 is a function that takes as input and poduces the output . 푓 : 푌 → 푋 푦 푥 .

[Example 3](continued) (d) For the function from [Example 3a]

function 푓: 푹 → 푹 defined by 푓(푥) = 4푥 − 5 does 푓 have an inverse function? If so, find the inverse function.

(d) For the function from [Example 3c]

function 푓: 풁 → 풁 defined by 푓(푛) = 4푛 − 5 does 푓 have an inverse function? If so, find the inverse function.

[Example 4](continued) (d) For the function from [Example 4c]

set 퐴 = 푹 − {3} = {푥 ∈ 푹|푥 ≠ 3} set 퐵 = 푹 − {1} = {푦 ∈ 푹|푦 ≠ 1} 푥−2 function defined by ( ) 푓: 퐴 → 퐵 푓 푥 = 푥−3 does 푓 have an inverse function? If so, find the inverse function.