Functions, Relations, and One-To-One Functions

functionrelationheaderfigOther than the concept of a number, I can’t think of a more important, pervasive, useful, and unifying mathematical concept than that of function and relation.

In a previous post, I explained how I introduce the concepts of functions and relations to my students in a concrete manner by using examples that illustrate how people use these concepts every day without realizing it. I have found this approach to be effective with beginning and advanced students. One of the important ideas that I want the reader to get from that post is that functions and relations are much more than some equation or formula such as y = f(x) = 2Sin(3x) – x,  y = g(x) = (x2-9)/x2 or |x| + |y| = 10. An equation is just one of many different types of matching rules of a relation. When a mathematician or scientist writes an equation, the equation is just the matching rule of some relation that tells us how to match domain elements with range elements. When a child is learning how to add two whole numbers, the child is learning how to use the addition function to match a pair of whole numbers with a single whole number. A diner in a restaurant uses the restaurant’s menu function to match food items with prices, determine what food items are in the restaurant’s domain, and the price range of the restaurant’s food items.

The purpose of this post is to discuss functions and relations when the matching rule is given by an x-y variable equation where both the domain and range is a subset of real numbers. Equations give us an algebraic description of the relation, and the graph of a relation gives us a geometric description of the relation.

I will begin by explaining the difference between a function and relation where the x-variable represents all possible numerical values in the domain of the relation, and the y-variable represents all possible numerical values in the range of the relation. All functions are relations, but not every relation is a function. For a relation to be a function of x, every value of x in the domain of the relation is matched with only one y value. The text box and the companion graph below give examples of relations where y is not a function of x. An ordered pair of real numbers (p, q) on the graph of a relation tells us that p matches with q. In each relation below, there is at least one x-value that matches with two or more different y-values which in turn tells us that there is a vertical line that intersects the graph of the relation in two or more points. All of these relations fail the vertical line test and therefore they are not functions of x. When the graph of a relation is given, it’s easy to tell whether or not the relation is also a function; just apply the vertical line test.

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I will now discuss the symbols that are used with functions. For beginning students, these symbols are very abstract. If a student never learns the meaning of these symbols, the student will never learn calculus. When we write an equation of the form y = f(x), we are saying that the variable y is an explicitly defined function of x such that every input value of the function matches with exactly one output value. The y variable represents an output value of the function, and is said to be a dependent variable because its value depends on the value of the x variable. The x variable represents an input value of the function, and is said to be an independent variable because its value can be any freely selected value in the domain of the function. The mathematical expression for the symbol f(x) tells us how to calculate the output value of the function for every x in the domain of the function. I routinely tell my students that the symbols y and f(x) represent the same thing, and the point (x, y) = (x, f(x)) tells us how high above or below the x-axis the point is.

Now let’s take a look at one-to-one functions. A function y = f(x) is one-to-one if and only if f(a) ≠ f(b) whenever a ≠ b. In other words, the output values of one-to-one functions are always different if the input values are different. A restaurant menu function is not one-to-one because different food items map to the same price. The Social Security function that matches people with valid social security numbers is one-to-one because two different people are always matched to different valid social security numbers. If a function is not one-to-one, there are two different numbers p and q in the domain such that f(p) = f(q) which in turn tells us there is a horizontal line that intersects the graph of the function in two or more points. When the graph of a relation is given, it’s easy to tell whether or not the relation is a one-to-one function. First apply the vertical line test to see if the relation is a function. If the relation is a function, apply the horizontal line test to see if the function is one-to-one.

The text box and companion graphs below describe a variety of explicitly defined functions of x. Notice that all of the one-to-one functions are either strictly increasing or strictly decreasing functions. If a function is even, it follows from the definition of an even function that the function is not a one-to-one function.

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The text box below shows how the concept of a one-to-one function can be used to solve a logarithmic equation. Step 3 in the solution follows from the fact that all Log functions are one-to-one. This type of reasoning is definitely a step up in mathematical maturity and sophistication for most students. Initially, some students think that both sides of the equation in step 2 were divided by Log. Of course, experienced math teachers know what I’m talking about. This reminds me of an old math joke. Question: What is sin x / n? Answer: sin x / n = six = 6.

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Useful tools from Math Teacher’s Resource

I will conclude this post by showing you the graph of a relation that only myself and some of my students have seen. The equation for this relation has nothing to do with anything. It’s just a somewhat random inequality that popped in my head. I show this relation to students to remind them that the graph of a mathematical relation can be any set of points in the x-y coordinate plane.

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