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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 = x²and 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

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. 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. Among these handouts are the Even and Odd Functions Handout and Relations and Functions Introduction handout. Go to mathteachersresource.com/instructional-content to download all the MTR handouts. All content is available for immediate download. No sign-up required; no strings attached!

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.

Solving Newton’s Law of Cooling/Heating Problems without Differential Calculus

GodfreyKneller-IsaacNewton-sm — Sir Isaac Newton (portrait by Godfrey Kneller, 1689)

My last post discussed how to find an exponential growth/decay equation that expresses a relationship between two variables by first constructing a table of data-pairs to better understand and derive the fundamental grow/decay equation A = A₀*b^t/k. Because the content of this post depends on the concepts developed in my last post, I strongly suggest that you read that post before continuing.

This post shows how to solve Newton’s law of cooling and heating problems without any understanding of differential calculus, which makes this post different from descriptions found in differential calculus text books. Newton’s Law of Cooling describes the relationship between the temperature of an object and time t when the object is placed in an environment where the ambient (or surrounding) temperature is maintained at a constant temperature. Newton’s law of cooling and heating is described as follows:

(a) If the initial temperature of the object equals the ambient temperature, the temperature of the object remains constant as time t increases.

(b) If the initial temperature of the object is greater than the ambient temperature, the object cools and its temperature exponentially and asymptotically approaches the ambient temperature as time t increases.

(c) If the initial temperature of the object is less than the ambient temperature, the object heats up and its temperature exponentially and asymptotically approaches the ambient temperature as time t increases.

I will use two familiar cooling/heating problems to illustrate how the table data-pair approach can be applied to solve a Newton’s law of cooling or heating problem. The key step in solving a cooling/heating problem is to carefully read the problem and then apply what Newton tells us about cooling and heating to create a rough sketch of the growth/decay graph of the model with key points labeled. Even if you don’t know the equation of the graph, the rough sketch will enable you to determine the parameters of the growth/decay equation. From this rough sketch, recognize that the graph is just the result of a vertical translation of an exponential decay graph in the form A = A₀*b^t/k. (In view of what Newton tells us about cooling and heating, the rough graph makes perfect sense to students.)

Problem 1: A pot of boiling soup is put into a sink filled with cold water. The temperature of the soup was 100⁰C when it was first put into the sink. By adding ice and stirring the water, the temperature of the water was maintained at a constant temperature of 5⁰C. If the temperature of the soup was 60⁰C after 10 minutes, how many minutes will it take for the temperature of the soup to reach a room temperature of 20⁰ C?

Solution: Refer to graphs A and B below where x = time t in minutes and y = the temperature of the soup in degrees Celsius. Similar to graph A, first draw a rough sketch of the model with key points labeled. Recognizing that graph A is just the result of a 5 unit vertical translation of an exponential decay graph, use the information from the first rough sketch to draw a rough sketch of the exponential decay graph with key points labeled, similar to graph B. Now use the key points on the sketch of graph B to find the equation of graph B, and then apply the equation transformation rules to find the equation of graph A. To find out how many minutes it will take for the temperature of the soup to reach 20⁰C, use a computer graphing program to find the intersection point of the graphs y = 20 and y = 95(55/95)^x/10 + 5. Graph A tells us the temperature of the soup equals 20⁰C, when time t = 33.77 minutes or about 34 minutes.

Problem 2: A 40⁰F roast is put into an oven that is set to bake at 350⁰F. After 2 hours, the temperature of the roast is 125⁰F. The roast is considered done when its internal temperature reaches 165⁰F. How many hours well it take to cook the roast?

Solution: Refer to graphs C and D below where x = time t in minutes and y = the temperature of the roast in degrees Fahrenheit. The strategy is to first draw a rough sketch of the model with key points labeled; similar to graph C below. Recognizing that graph C is the result of a 350 unit vertical translation of an exponential decay graph that was reflected over the x-axis, use the information from the sketch of graph C to draw a rough sketch of the flipped exponential decay graph with key points labeled, similar to graph D. Now use the key points on the sketch of graph D to find the equation of Graph D, and then apply the equation transformation rules to find the equation of graph C. To find out how many hours it will take to cook the roast, use a computer graphing program to find the intersection point of the graphs y = 165 and y = -310(225/310)^x/2 + 350. Graph C tells us that it will take 3.22 hours or about 3 hours and 13 minutes to cook the roast.

Here are four exercises that you can give to your students. The solutions are provided. (See my third comment below.) You or your students shouldn’t be too disappointed if you fail to correctly solve all four exercises on your first attempt.

Exercise 1: When first removed from an oven and placed in a 70⁰F room to cool, the temperature of a cake was 180⁰F. Three minutes later the temperature of the cake dropped to 160⁰F.

(a) What is the temperature of the cake after 20 minutes? (A: 98.87⁰F or about 99⁰F)
(b) How many minutes will take for the cake to cool to 90⁰F? (A: 25.49 minutes or about 26 minutes)

Exercise 2: The temperature of a very small metal bar was 30⁰C when it was dropped into a large barrel of hot water having a 75⁰C temperature. After 1 second, the temperature of the bar was 31⁰C.

(a) How long will it take for the temperature of the bar to reach 70⁰C? (A: 97.77 seconds or about 98 seconds)
(b) How long will it take for the temperature of the bar to reach 74⁰C? (A: 169.39 seconds or about 170 seconds)

Exercise 3: Find the equation of graph A below. (A: y = 40(1/5)^x/10 + 30)

Exercise 4: Find the equation of graph B below. (A: y = -90(2/3)^x/5 + 160)

newton_fig5a

Comments:

• In the two sample problems above, the final step in the solution involved finding the intersection point of two graphs. This gives us the solution from a geometric point of view. The solution from an algebraic point of view involves log functions which would enable you to find the solution faster. As I mentioned in previous posts, whenever possible, solutions to problems should be understood from both an algebraic and geometric point of view.

• Solving exponential growth and decay problems naturally leads to a need to understand logarithms and log functions.

• All modern physicists know that the equations they discovered can only give us an approximation of how nature’s laws work. In reference to problem (2) above, if we conducted an experiment with a roast by measuring its internal temperature at various points in time, we would find a discrepancy between the experimental results and the predicted results. No matter how accurately we measure the internal temperature of the roast and time, the errors can’t be taken out of the experimental observations. We can only say that the interval temperate of the roast at some specific point in time lies in an area of uncertainly which is the area under a probability distribution curve. This is why least-squares regression equations are used to describe the relationship between two variables.

• I have used the handout Newton’s Law of Cooling with college algebra and pre-calculus students, and with more advanced students that I tutor. To download the free student and teacher versions of the handout, go to mathteachersresource.com/instructional-content. There are other free handouts on properties of exponents, properties of logarithms, solving exponential/logarithmic equations, and logarithmic base conversion.

• Using the approach presented in my last post and this post, I believe it’s possible to teach how to solve exponential growth/decay problems to younger mathematically capable students. From my own experience, students find these types of problems interesting and practical.

• All graphs in this post were created with my program Basic Trig Functions. I designed the program to make it easy for teachers to create content for their own courses.

My next post will discuss the derivation of the formula for the future value of an investment when interest is compound continuously, FV = Pe^rt. The post will assume that the reader has no understanding of the limit concept in calculus.

Exponential Growth and Decay from a Data-Pairs Approach

My last two posts discussed the mathematics of linear growth and decay. If you have not read those posts, you might find it helpful to read them before continuing. This post focuses on finding an exponential equation that expresses a relationship between two variables by first constructing a table of data-pairs to better understand the relationship and see the pattern in the relationship.

Most exponential growth/decay relationships involve a time variable t and the amount A of some quantity at time t. Amount could be the current value of an investment account, population of a city, remaining kilograms of radioactive material, assessed value of a truck, etc. The text box and observations below explain how and why the basic fundamental exponential growth/decay formula A = A₀*b^t/k works, and the role that the parameters A₀, b, and k play in the equation. Periodic growth factor is another way to think of the base multiplier b.

Some observations about A = A₀*b^t/kwhere b > 0:
• The point (0, A₀) is the intercept on the vertical axis of the graph.
• Base multiplier b is a periodic growth or decay factor.
• If 0 < b < 1, the equation models exponential decay.
• If b > 1, the equation models exponential growth.
• Exponential growth/decay is about repeated multiplication by growth/decay factor b.
• A₀ and any other point on the graph determines a unique exponential equation.
• If A₀ is positive, the graph is above and asymptotic to the horizontal axis.
• If A₀ is negative, the graph is below and asymptotic to the horizontal axis.

Most discussions about finding the equation of an exponential relationship don’t start by looking at data-pairs in a table. After only a couple of demonstrations of how to apply the data-pairs approach, students quickly develop the ability to find the three key parameters of an exponential growth/decay relationship. Exponential equations of the form A = A₀*b^t/k where base b is a rational number are much easier to comprehend than equations of the form A = A₀*e^ktwhere e is the irrational math constant = 2.718281828459045 . . . . I will use four familiar math problems that involve an exponential relationship to illustrate the table data-pairs approach. In the comments section of this post, you will find an example that further clarifies my reason for expressing most exponential growth/decay equations as A = A₀*b^t/kwhere b is a rational number, and the reason that the solutions of population and radioactive growth/decay problems tend to be expressed in terms of base e only.

Problem 1: Consider a population of bacteria that is growing exponentially 50% every 4 hours and the current population is 60 bacteria. Let t = number of hours in the future and N = the number of bacteria after t hours.
(a) Find an equation that expresses N as a function of t.
(b) Find the population after 10 hours and 45 minutes ago.
(c) Express N as a function of t if the population is increasing 5% every 15 minutes.

The solution is given in the text box below. Problem solvers should carefully read the problem, create a table of data-pairs, determine the equation parameters, and then write the equation that models the problem situation. A companion exponential growth graph with a series of slope/rate triangles is provided to show the role that the equation parameters play in the relationship. Of course, the problem solver should always check the solution by using a computer graphing program to graph the equation.

Problem 2: Suppose a person invests $10,000 in a CD that will earn interest at 6%/year and interest is compounded monthly. Let t = the number of years in the future and V = the value of the investment after t years.
(a) Express V as a function of t.
(b) Find the value of the investment after 10 years and 20 years.
(c) Express V as a function of t if interest is compounded 360 times per year.

Problem 3: The half-life of a radioactive substance equals the time it takes (20 days, 149 years, 5,700 years, etc.) for the substance to lose half its mass. Consider a radioactive substance with a half-life of 60 days that currently has a 100 kg mass. Let time t = the number of days in the future and A = the mass of the remaining substance in kg at time t. Refer to table and companion graph below.

(a) Find a formula that expresses A as a function of time t in days.
(b) Find the mass of the substance after 135 days.
(c) Find a formula for A(t) if the half-life = 6 hours instead of 60 days.

Problem 4: The two exponential growth/decay graphs along with key points on the graphs are shown below.
(a) For graph A: Write an equation that expresses y as a function of x.
(b) For graph B: Write an equation that expresses y as a function of x.

Here are four exercises that you can give to your students. The graphs are a mixture of linear and exponential growth/decay graphs. Using the points on the graph, find the equation of the graph. If you wish, remind them that they should first create a table of data-pairs. Let them do the exercises with a partner and then check their answers by using a computer to graph the equations. We want to create a save environment in which kids feel free to experiment and check their answers for understanding. It’s OK to make a mistake, just fix it. If the first attempt to fix a mistake fails, so what? Try again. This is how real people learn to do anything that is worthwhile. The solutions are given at the end of this post.

Comments:
• Consider the two mathematically equivalent equations below that model the population growth of a small town where t equals the number of years after 2010.

P = 5,200(1.08)^t/4 and P = 5,200e^0.019240260t

The first equation immediately tells us the population of the town was 5,200 in 2010, and the population is increasing 8% every 4 years. The second equation tells us the population of the town was 5,200 in 2010, but by just inspecting the second equation, only God can figure out that the population is increasing 8% every four years. (Increasing 8% every 4 years is slightly less than increasing 2% every year.)

• It’s a snap to find the derivative of functions of the form y = Ae^kt. To find the derivative of functions of the form y = Ab^x/k where base b is a rational number requires a little more work. I suspect this is the reason that the solutions of population and radioactive grow/decay problems tend to be expressed in terms of base e only. From my point of view, this is not a sufficient reason to do so because converting an exponential function from one base to another base is a simple procedure. My free handout Logarithmic Base Conversion shows how to do this.

• All modern physicists know that the equations they discovered can only give us an approximation of how nature’s laws work. The brilliant physicist Richard Feynman, over and over again, stated this fundamental fact in his lectures and talks. In reference to problem (3) above, if we conducted an experiment with a radioactive material by measuring the remaining mass of the material at various points in time, we would find a discrepancy between the experimental results and the predicted results. No matter how accurately we measure mass and time, the errors can’t be taken out of the experimental observations. We can only say that the remaining mass of radioactive material at time t lies in an area of uncertainly which is the area under a probability distribution curve. This is why least-squares regression equations are used to describe the relationship between two variables.

• The formula for calculating the future value of an account after t years when interest is compounded continuously is FV = Pe^rt where P = the principal and r = the annual interest rate expressed as a decimal. It’s impossible to express this relationship with a base that is a rational number. In a future post, I will give a derivation of this formula in a manner that does not require an understanding of concepts in calculus.

• I have used the handout, Introduction to Exponential Growth and Decay, with college algebra students, pre-calculus students, and as a review for more advanced students. To download the free student and teacher versions of the handout, go to mathteachersresource.com/instructional-content.html. There are other free handouts on properties of exponents, properties of logarithms, solving exponential/logarithmic equations, and logarithmic base conversion.

• All graphs in this post were created with my software, Basic Trig Functions. I designed the software to help teachers quickly make custom content for their classrooms. This software allows you to easily copy any graphic and then import it directly into a document (e.g. lesson plan, class handout, test) or further manipulate it in various graphic processing programs.

My next post will show how to solve Newton’s Law of Cooling problems without understanding differential calculus.

Solutions to exercises:
Graph A: y = -2x + 40
Graph B: y = 40*0.5^x/2.5
Graph C: y = 20*1.5^x/5
Graph D: y = x + 10