## Inverses of Functions and Relations

My previous post discussed the mathematical concepts of function and relation. Because the content of this post heavily depends on an understanding of the ideas presented in that post, you may find it helpful to read it before continuing.

The concept of the inverse of a relation is a natural extension of the important concept of a relation. The central idea is that an inverse relation is about reversing a relationship by exchanging variables, reversing/undoing an operation, or reversing/undoing a series of operations in a specific order. The following five questions and situations illustrate how a person uses the concept of the inverse of a relation to solve a problem.

(1) If we know a formula to convert Fahrenheit temperatures to Celsius, what formula converts Celsius temperatures to Fahrenheit?

(2) If we know a formula that tells us how to calculate the area of a circle from its radius, what formula will tell us how to calculate the radius of the circle from its area?

(3) A diner in a restaurant uses the restaurant’s menu function in inverse mode to determine what food items on the menu he/she can afford.

(4) A criminal investigator uses the one-to-one function that matches people with DNA molecules in inverse mode to match a sample of DNA molecules with a criminal.

(5) When solving for the sides and angles of a triangle, a trig student uses the inverse trig functions on his/her calculator to find the measure of an angle that has a specific trig function value.

The purpose of this post is to discuss inverse 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. These concepts will be discussed from algebraic and geometric points of view.

I will begin by looking at inverses of functions and relations from a geometric point of view. The two text boxes below summarize the geometric relationships between a relation and the inverse of a relation. The companion graphs illustrate the geometric relationships described in the text boxes. Notice that exchanging the variables in an equation gives us the equation of the inverse relation. These observations, of course, follow from the definition of the inverse of a relation, midpoint formula, definition of slope, and the fact that the product of the slopes of two perpendicular lines equals -1.

The text box below shows examples of elementary functions and the corresponding inverse relation which may or may not be a function. Notice that the inverse of the functions y = x2 and y = |x| are relations, but not functions since y = x2 and y = |x| are not one-to-one functions. As a reminder, the symbol √(x) means take the positive square root of x, and positive real numbers have a positive square root and a negative square root. Also note that the function y = Sin(x) is not one-to-one, and therefore the inverse relation is not a function. Calculators get around this problem by restricting the range of the function Sin-1(x) to values that range from –π/2 to π/2.

The next part explains how I teach the inverse of trig functions y = Sin(x) and y = Cos(x). Initially, students struggle with the definitions of the inverse trig functions. Consider the equations listed in the edit box and graphs below. Because the trig functions are periodic, there are infinitely many solutions for each equation. Because the calculator keys Cos-1(x) and Sin-1(x) are function keys, the calculator should display only one of the infinitely possible output values. When x ranges from 0 to π, Cos(x) is one-to-one in adjacent quadrants I and II, and all possible output values of Cos(x) from -1 to 1 can be generated in quadrants I and II. Therefore Cos-1(x) is a function if the output is restricted to range values from 0 to π radians. When x ranges from – π/2 to π/2, Sin(x) is one-to-one in adjacent quadrants I and IV, and all possible output values of Sin(x) from -1 to 1 can be generated in quadrants I and IV. Therefore Sin-1(x) is a function if the output is restricted to range values from – π/2 to π/2 radians.  I have my trig students find six solutions of simple trig equations.  Example: Find six angles β in degrees in quadrant III, 3 positive and 3 negative, such that Cos(β) = -0.951056516. Round solutions to the nearest tenth of a degree.

I will conclude this post my showing you how I teach my students to find the inverse of a function when the function is composed of basic functions. The steps in the algorithm involve applying inverse operations in the reverse order of the order of operation rules. Exercises of this type reinforce concepts and are a good way to practice algebra skills. If you want to add some rigor to your course, have students check their solution by showing f(f-1(x)) = f -1(f(x)) = x. I remind students that an initial equation like x = y/(3y – 4) is an equation of the inverse relation, but it’s not expressed as a function of x. When a relationship is expressed as a function of x, we can graph the relation with a graphing utility. This is one of the reasons that we teach kids to solve an equation for a given variable. Sometimes I tell students to rearrange the equation for some variable because it makes more sense to them.

Useful tools from Math Teacher’s Resource:

•   The graphs in my posts are created with my software, Basic Trig Functions. I think that you will find it very useful for teaching mathematical concepts in your classroom and developing custom instructional content. Relations can be entered as an explicitly defined function of x, an explicitly defined function of y, or as an implicitly defined x-y variable relation. Check it out at mathteachersresource.com/trigonometry.

•   There are a wide variety of free handouts that teachers can use to create lessons or give to students as a handy reference handout. Among these handouts are Inverse Relations and Functions, Even and Odd Functions, and Relations and Functions Introduction handouts. Go to mathteachersresource.com/instructional-content to download MTR handouts. All content is available for immediate download. No sign-up required; no strings attached!

Comments Regarding My Previous Post:

•   Some readers wanted to know the equation of the lead graph in my previous post. The equation of the graph is Cos(x) + Cos(y) >= 0.4 where both x and y range from -15 to 15. In view of the fact that Cos(x) is an even function, it should be no surprise that the graph has symmetry with respect to the x-axis, the y-axis, and the origin.

•   The equation of the strange graph at the end of my previous post is 2xSin(3x) + 2y <= 3yCos(x + 2y) + 1. If you are skeptical, here are six solutions that you can plug into the equation to verify that the equation really does have solutions that satisfy the equality relationship. Just make sure that your calculator is in radian angle mode.

(-5.4, 5.195 577 636)

(5.5, 5.976 946 313)

(8.680 865 276, -5.2)

(-6.8, -6.786 215 284)

(0.578 827 17, -3)

(0.051 781 64, 5.8)

## Functions, Relations, and One-To-One Functions

Other 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.

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.

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.

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.

## Modeling Limited Population Growth with the Logistic Function

Because of limits on food, living space, disease, current technology, war, and other factors, most populations have limited growth as opposed to unlimited exponential growth which is modeled by the classic exponential growth equation P = P0bt/k. A limited growth population starts growing almost exponentially, but reaches a critical point in time where its growth rate slows, and the population starts to asymptotically approach an upper limit as time increases. There are several models that are used to describe limited growth of a population.

In this post, I will discuss the logistic function which was used by the Belgian mathematician Pierre Francois Verhulst (1804-1849) to study limited population growth. The logistic function also has applications in artificial neural networks, biology, chemistry, demography, ecology, economics, biomathematics, geoscience, mathematical psychology, sociology, political science, probability, and statistics.

The two text boxes below describes the key parameters and relationships between the parameters of a logistic function. Graph (A) shows a typical logistic function curve and how equation parameters can be calculated from known characteristics of the population. If pmax, p0, and tc are known, then a, b, and k can be calculated. Likewise, if a, b, and k are known, then pmax, p0, and tc can be calculated. Keep in mind that all limited growth models can only give us a good approximation of a population value at some point in time.

In most cases, the key parameters of a logistic equation are unknown, but an observed set of data-pairs is known. The least-squares logistic equation of a data set is the best of all possible logistic equations that describes the relationship between the data-pair variables. Best possible equation means that the sum of the squared errors (difference between observed value and predicted value) is minimized. Modern graphing calculators have the capability of findings a least-squares equation for a variety of models such as linear, quadratic, cubic, quartic, sinusoidal, log, exponential, and logistic. When given a logistic type data set, I will use a graphing calculator to find the least-squares logistic equation of the data set, and then calculate various characteristics and properties of the resulting logistic model. I will now take a look at two problems that illustrate how the logistic function can be used to describe limited population growth.

Problem 1 solution: Use math software to do a scatter plot of the data, find the least-squares logistic equation p = 12.0121 / (1 + 10.6694e-0.023856x) of the data set, and then do the appropriate calculations. Refer to graph (B) below. From graph (B) we see that the world population growth rate started to slow in 1999, and the upper limit of the world population is about 12 billion. Keep in mind that this least-squares equation is our current best description of world population growth. Future unknowable events will alter this model.

Problem 2: The logistic function N(t) = 3,600 / (1 + 29.4e-0.2t ) models the spread of a disease in a town. N(t) = the total number of people infected at time t, and t = the number of days after the first reported infections.

(a) How many people were initially infected?

(b) How many people were infected after 10 days and after 30 days?

(c) When did the rate of infection start to slow?

(d) What is the upper limit of the number of infected people?

Problem 2 solution: Use math software to graph the equation, and then do the appropriate calculations. Refer to graph (C) below.

(a) About 118 people were initially infected.

(b) After 10 and 30 days, 723 and 3,355 were infected.

(c) About day 17, the infection rate started to slow.

(d) The upper limit of the number of people infected = 3,600.