MA.912.F.3.4

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.AR.5.8
• MA.912.AR.5.9

Terms from the K-12 Glossary

• Domain 
• Function 
• Function notation

Vertical Alignment

Previous Benchmarks

Next Benchmarks

• MA.912.C.2.3
• MA.912.C.2.4

Purpose and Instructional Strategies

• Define function composition and demonstrate composing functions with several examples. Explain that function composition is an operation that can be performed on two functions so the output of one function becomes the input of another to form a new function. The resulting function is known as a composite function. 
• Introduce the notation for composition of functions as ($f$ _°$\mathrm{ g}$)($x$) = $f$($g$($x$)) and explain that we read the left-hand side as “$f$ composed with $g$ at $x$,” and the right-hand side as “$f$ of $g$ of $x$.” 
• Emphasize the difference between $f$($g$($x$)) and $g$($f$($x$)) and explain that function composition is not commutative. Explain that it is important to follow the order of operations when evaluating a composite function, we evaluate the inner function first. 
• Instruction includes determining the domain and range of the composite function. Explain that the domain of the composite function is all inputs $x$, such that $x$ is in the domain of $g$ and $g$($x$) is in the domain of $f$.

Common Misconceptions or Errors

• Students sometimes believe that variables represent just a fixed number, so they struggle understanding that a variable can also represent a function.
• Students struggle evaluating functions for numerical inputs, so it is more difficult for them evaluating for inputs that are functions. 
• Students may not understand the function notation for composition thinking that $f$($g$($x$)) means to multiply function $g$ by function $f$. 
• Students confuse the composition ($f$_° $g$) with the product ($f$ · $g$). Emphasize that the composition means “evaluate $g$ at $x$, then evaluate $f$ at the result $g$($x$).” 
• Some students might thing that ($f$ _° $g$) is the same as ($g$ _°$\mathrm{ f}$). Emphasize that composition is not commutative. 
• When finding the domain of a composite function, sometimes students find the union of the domains instead of the intersection. Emphasize that they need to find what the domains have in common, the intersection. Also, students sometimes forget to include the restrictions on the domain of the inner function on the domain of the composite function. Remind them to use the most restrictive domain.

Instructional Tasks

• Your parents purchased a new reclining chair for the living room. The chair will be delivered to your home. The cost of the chair is $d$ dollars, the tax rate is 6.5%, and the delivery fee is $50. 
  • Part A. Write a function $f$($d$) for the purchase amount, $x$, and the delivery fee. State the domain. 
  • Part B. Write another function $g$($d$) for the cost of the chair after taxes. State the domain. 
  • Part C. Write the function $f$($g$($d$)) and interpret its meaning. State the domain of $f$($g$($d$)). 
  • Part D. Write the function $g$($f$($d$)) and interpret its meaning. State the domain of $g$($f$($d$)). 
  • Part E. Which results in a lower cost to you, $f$($g$($x$)) or $g$($f$($x$))? Explain why. 

• Let $f$ be the function that assigns to a temperature in degrees Celsius its equivalent in degrees Fahrenheit. Let $g$ be the function that assigns to a temperature in degrees Kelvin its equivalent in degrees Celsius. 
  • Part A. Explain what $x$ and $f$($g$($x$)) represent in terms of temperatures, or explain why there is no reasonable representation. 
  • Part B. Explain what $x$ and $g$($f$($x$)) represent in terms of temperatures, or explain why there is no reasonable representation.  
  • Part C. Given that $f$($x$) = $\frac{\text{9}}{\text{5}}x$ + 32 and $g$($x$) = $x$ − 273, find an expression for $f$($g$($x$)). 
  • Part D. Find an expression for the function $h$ which assigns to a temperature in degrees Fahrenheit its equivalent in degrees Kelvin.

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