Repeatedly solve equations of the form f(x)=c to recognize the need for a function that "undoes" the original function, i.e. to find the value in the domain that generates a certain output.
Understand the relationship between the inputs and outputs of a function and its inverse and use this to evaluate inverse functions.
Find an inverse function algebraically.
Verify by composition that one function is the inverse of another.
Quick Lesson Plan
In this lesson students consider how their puppy’s diet determines their adult weight, and inversely, how an ideal adult weight determines how much the dog should be fed as a puppy.
Question 1c is purposefully straightforward, allowing students to read values of an inverse function from a table). Although students may not yet see this question as being related to inverses, they establish a rudimentary idea of how the relationship between diet and adult weight can be seen in two ways depending on what information is given and which variable is the independent or dependent variable.
Questions 2 and 3 have students use identical lines of reasoning to begin to make use of structure. Students should see that in order to solve the equation 30=(k/130)^(4/3), they must first raise both sides to the (¾) power and then multiply by 130. This repeated structure helps students come up with the inverse equation and realize that any time we are given a weight w and want to determine the number of calories to feed the puppy, we must raise the weight to the (¾) power and multiply by 130.
Question 5 should be intuitive for students: the idea that they can put their calculated calorie count back into the original equation to see if it will give the desired weight. This is the underlying principle behind defining inverses in terms of compositions and it’s important that students see this conceptually before they are asked to give algebraic proofs.
What is the input of this function? What is the output?
How are you determining how many calories the dog should be fed?
How do you undo something raised to the three-fourths power?
If I give you a weight, can you give me the kilocalories? Which equation would you use? Which equation could you use to check that that number of kilocalories will actually produce the desired weight?
While the idea of inverses is pretty intuitive (we undo processes all the time), the algebraic component can provide many challenges for students. Much of the confusion can be attributed to the notation and the vocabulary used with inverses. In this lesson we specifically hold off on formal function notation until the debrief, when we believe students will have the conceptual understanding to make sense of the symbolic representation.
The biggest challenge with inverse notation comes with the problematic use of the “switch x and y” procedure. This is because very rarely do students notice that the x and y in f^-1(x)=y actually represent different things than they did in the original function. In our margin notes, we use the notation f^-1(w)=k to denote that the inverse function inputs weight and outputs kilocalories which they have already seen in the table and their equations in questions 2-4. Furthermore, we use the language of “solve for the dependent variable” to replace the traditional “switch x and y”. Using the dog example, since kilocalories, k, is the dependent variable of the inverse function, we need to isolate this variable. We can then replace k with f^-1(w) since k is a function in terms of w. This shift in language certainly requires an adjustment, especially since many of our textbooks use the more problematic language, but we believe it will be worthwhile in helping students make sense of inverse functions. For a more in depth discussion of this principle, check out this article from the American Mathematical Society.