Inverse functions
• Solve for $x$ such that $f(x) = c$, when $f$ is a linear function (F-BF.B.4a).
• Write an expression for an inverse (F-BF.4a).
In this section, students learn about inverse functions. The tasks illustrate some real-world contexts in which inverse functions occur.
Tasks
1
Your Father
WHAT: $F$ is a rule that assigns a student in the mathematics class to his or her biological father. Students are asked to: explain why $F$ is a function; describe conditions on the class that ensure F has an inverse; and, in a case where $F$ does not have an inverse, how to modify the domain of F to ensure it does.
WHY: This task revisits the definition of function in a situation where domain and range are not sets of numbers, and where it’s easy to state the rules for $F$ and its inverse, and to describe an appropriate restriction of the domain.
2
Temperatures in degrees Fahrenheit and Celsius
WHAT: Given Fahrenheit and Celsius measurements for the temperatures of boiling and freezing water, students construct a linear function that converts measurements in degrees Fahrenheit to those in Celsius. Students then find the inverse of this function and explain its meaning in terms of temperature conversion. Lastly, students are asked if there is a temperature which is the same in Fahrenheit and Celsius.
WHY: Students use their knowledge of linear functions to construct a function in a real-world context, and to see one example of how a measurement conversion can be viewed as a linear function or its inverse. The last part of the task is an opportunity to work with the relationship between the two scales, using the expression for one of the conversion functions, or using pairs of equivalent temperatures as described in the task commentary.
3
US Households
WHAT: This task presents a table that gives the number of households in the US between 1998 and 2004. Students are asked to find a linear function h that approximates the number of households (in thousands) as a function of the year(MP4). Students then write an expression for h–1 and interpret $h^{–1}(111,000)$ in terms of number of households.
WHY: This task provides an opportunity for students construct a function and its inverse in a real-world context, and to interpret a value of the inverse in terms of the quantity represented, finding a referent for $h^{–1}(111,000)$ (MP2). Because this task uses notation for the inverse of a function, it can also serve as an introduction to this notation.
External Resources
1
Noticing features of inverse functions
Description
WHAT: This post describes two days of instruction. First, students approach the idea of an inverse operation in the context of ciphers. They then write rules by listing out operations for a given function and then writing a new expression which “undoes” those operations (F-BF.B.4a). The next day, students create tables and graphs for functions and their inverses. They look for commonalities as they consider pairs of functions and their inverses. Note: This lesson was created for an Algebra 2 class, so depending on students’ prior experience with various function types, it may need to be modified to make it appropriate for a given classroom.
WHY: This lesson highlights fundamental characteristics about functions and their inverses: that it can generally be thought of as a process of “undoing,” and also that the inputs and outputs trade places. This lesson does not address the idea that a function only has an inverse function if it is one to one, but that idea is addressed in later activities in this section.