Linear Functions and Applications

1. Graph a contextual function and its inverse on a coordinate plane. 
2. Define that inverse functions are related graphically by a reflection over the line $${y = x}$$. 
3. Describe the meaning of the original function and inverse function in terms of the context. 
4. Represent a contextual situation and its inverse in tables of values. 
5. Describe the meaning of the input or  in each function and its inverse. 
6. Use inverse notation, $${f^{-1}(x)}$$ to describe an inverse function. 

• Inverse functions are taught through a linear lens in Algebra 1 and reviewed in this lesson. Inverse functions are very important in AP Calculus, so additional time is spent to ensure that students have a solid base in the conceptual understanding. 
• In future lessons, students will verify inverse by composition, but this is not a criterion for success for this lesson and the next lesson. 
• This lesson has components that extend beyond F-BF.4a into F-BF.4c and F-BF.4d. If you are not teaching an advanced Algebra 2 course, focus on the contextual meaning of inverse functions presented in this lesson rather than the tabular or graphical analysis of inverse functions. 
• The following resource may be helpful for teachers to grasp the full conceptual understanding of inverse functions before planning this lesson. American Mathematical Society Blogs, Art Duval, “Inverse Functions: We’re Teaching It All Wrong!” November 28, 2016.

Below is a graph that represents $${f(x) = 0.2x}$$ where $$x$$ represents the number of steps taken and $$f(x)$$ represents the number of calories burned for a person weighing 160 pounds. 

• Fill in the table of values for both the function and the inverse for the first few values in the domain. 
• Graph the inverse function. 
• Label the axes in both graphs according to the context of the problem. 
• Describe the function and its inverse in context.