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)