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.