MA.912.F.3.7

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.AR.5.6
• MA.912.AR.5.7
• MA.912.AR.5.8
• MA.912.AR.5.9 
• MA.912.F.1.6

Terms from the K-12 Glossary

• Inverse functions

Vertical Alignment

Previous Benchmarks

Next Benchmarks

• MA.912.F.3.8
• MA.912.F.3.9 
• MA.912.C.2.6

Purpose and Instructional Strategies

• Instruction includes noticing patterns such as in transposing of the input and output values in a table to create an inverse of a function. 
• Instruction includes the understanding that an inverse to a function does not always represent a function. Students should be able to know how to look at two functions in any representation and determine if they are inverse functions. 
• Instruction includes the connection to MA.912.F.3.6 to help students be able to represent inverses to given functions, if they exist, algebraically, graphically or in a table. If students are not able to determine whether an inverse to a given function exists be examining the features given in the function, they may struggle to be able to represent the inverse of the given function in various representations. In this benchmark, student are also expected to use compositions of functions to verify that one function is the inverse of the other. 
• Students need to show understanding that certain function types are inverses of each other such as quadratic, exponential and logarithmic functions. Students should also demonstrate an understanding that a linear function will have a linear inverse, if the inverse of the function exists (MTR.5.1).
• Instruction includes giving students a function represented in a table, graph, or equation and asking them to determine if the function is invertible (able to produce an inverse function) or non-invertible (not able to produce an inverse function). Once they have determined if the inverse function is able to be produced, then students should be expected to represent that inverse in multiple ways. In a table, students should notice that the input and output values transpose with each other. In a graph, students should notice that the points reflect over the line $y$ = $x$. 
• Students should be able to use compositions of function to verity that two functions are inverses of each other. When they compose the functions, they should demonstrate an understanding that if they end up with anything other than $x$ as their solution to the compositions of functions than those functions are not inverses of each other. 
• During instruction teachers may ask the students to use another representation (table or graph) to look at the two functions to justify their response as to whether the functions are inverses of each other. If they can see graphically that two functions are inverses of each other but their algebraically they do not get an $x$ when composing the functions, it will prompt the students to go back and check their work. Likewise, if they see graphically or in a table that the two functions are not inverses of each other, but their composition of functions does produce an $x$ they will be prompted to go back and check their algebra (MTR 6.1). 
• Instruction includes opportunities for students to explore the idea that when functions are inverses of each other, they can compose the functions both ways [$f$($g$($x$)) and $g$($f$($x$))] and still get “$x$” as the solution. While compositions are not commutative, in this special case, compositions can be done in either order.

Common Misconceptions or Errors

• Students may think that they should get a numerical value (most commonly zero or 1) for their solution when composing the two functions in order for the functions to be inverses, rather than getting the solution “$x$.” 
• Students may not understand how the graphs of inverse functions will reflect over the line $y$ = $x$. Students may not realize that it has to reflect over the line $y$ = $x$ rather than just be a reflection over any line. 
• Students may not understand how to look for patterns in the table that will justify that a function is an inverse of their given function.

Instructional Tasks

• The table below shows the number of households in the U.S. in the years. 
  
  
  

• Part A. Find a linear function, $h$, which models the number of households in the U.S. (in thousands) as a function of the year, $t$. 
• Part B. Represent the inverse of the linear function you wrote in Part A in a table, graph, and expression. 

• Find the inverse to the function $f$($x$) = $x$² + 3.
  a. Represent the inverse in an equation/expression, a table, and a graph.
  b. Verify that the inverse you found was actually an inverse algebraically using compositions of functions.

• Find the inverse of the function $f$($x$) = 3^$x$

• Find the inverse of the function represented in the table. Represent the inverse in a table. 

• Verify that the following functions are inverses of each other using compositions of functions.
  
  $f$($x$) = 2^$x$
  $g$($x$) = log₂$x$

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