The purpose of this task is to develop and extend the concept of inverse functions in a linear context. In the task, students use tables, graphs, and equations to represent inverse functions as two different ways of modeling the same situation. The representations expose the idea that the domain of the function is the range of the inverse (and vice versa) for suitably restricted domains. Students may also notice that the graphs of the inverse functions are reflections over the ! = $ line, but with the understanding that the axes are part of the reflection.

The purpose of this task is to extend students’ understanding of inverse functions to include quadratic functions and square roots. In the task, students are given the equation of a quadratic function for braking distance and asked to think about the distance needed to stop safely for a given speed. The logic is then turned around and students are asked to model the speed the car was going for a given braking distance. After students have worked with the inverse relationships with the limited domain [0, 217] (assuming that 217 is the maximum speed of the car), then students are asked to consider the quadratic function and its inverse on the entire domain. This is to elicit a discussion of when and under what conditions the inverse of a function is also a function.

Sample Learning Goals
Define a function as a rule relating each input to exactly one output and predictably acting on inputs
Predict outputs of a function using given inputs
Compose functions to create a new function
Determine which functions are geometric transformations

This worksheet covers finding inverses, graphing them and determining if one function is the inverse of the other.

This worksheet covers finding inverses, graphing them and determining if one function is the inverse of the other. Also have a critical thinking component about inverses

In this task students solidify their understanding of using trigonometric functions to model periodic behavior by applying trigonometry to a context that is periodic (high and low tides), but not circular motion. They learn how to interpret amplitude and period in terms of this new context. They also consider using inverse trigonometric functions to answer questions about the time at which the tide reaches various heights. Students answer these inverse questions both graphically and algebraically.

In this webpage, students learn what the inverse of a function is, and how to evaluate inverses of functions that are given in tables or graphs.

The purpose of this task is for students to become more fluent in finding inverses and to increase their flexibility in thinking about inverse functions using tables, graphs, equations, and verbal descriptions.

Interactive tool that allows one to see how the inverse of y=mx+b changes when the slope changes.

In this lesson, students will learn how to represent functions and their inverses in multiple ways, make meaningful interpretations of the inverse of a function and use them in real-world applications.

This worksheet covers finding the inverse functions for different logarithm functions and exponentials

The purpose of this task is to solidify students’ understanding of the relationship between functions and their inverses and to formalize writing inverse functions. In the task, students are given a function and a particular value for input value #, and then asked to describe and write the function that that will produce an output that is the original # value. The task relies on students’ intuitive understanding of inverse operations such as subtraction “undoing” addition or square roots “undoing” squaring. There are two exponential problems where students can describe “undoing” an exponential function and the teacher can support the writing of the inverse function using logarithmic notation.

The purpose of this task is two-fold:
1. To extend the ideas of inverse to exponential functions.
2. To develop informal ideas about logarithms based upon understanding inverses. (These ideas will be extended and formalized in Module II: Logarithmic Functions)

The first part of the task builds on earlier work in Secondary I and II with exponential functions. Students are given an exponential context, the distance travelled for a given time, and then asked to reverse their thinking to consider the time travelled for a given distance. After creating a graph of the situation, students are asked to consider the extended domain of all real numbers and think about how that affects the inverse function. Questions at the end of the task press on understanding of the inverse relationship and the idea that a function and its inverse “undo” each other, using function notation and particular values of a function and its inverse.