Getting Started 
Misconception/Error The student is unable to correctly write functions that model the given relationships. 
Examples of Student Work at this Level The student is unable to write the functions f(x), g(x), f(g(x)), and g(f(x)). For example, the student writes:
 For f(x):
 f(x) = 0.065x + 50
 f(x) = x + 6.5 + 50
 f(6.5) = 50(6.5)
 For g(x):
 g(x)= x + 50.065
 50g(x) = 0.065x
 g(x) = x + 6.5
 g(x) = 0.65x
 g(x) = 50 + x + 0.065x
The student is unable to correctly compose functions f(g(x)) and g(f(x)).

Questions Eliciting Thinking What does the variable x represent? Can you describe in words what function f represents? Can you describe in words what function g represents? How is tax on a purchase determined? Do you know what f(g(x)) means? How about g(f(x))? 
Instructional Implications Help the student identify the relevant variables and quantities given in the problem and then verbally describe their relationship. Guide the student to translate his or her verbal description into an equation using the notation given in the problem. Remind the student that calculating the cost, including taxes, requires the tax amount to be determined first and then added to the cost. Demonstrate the calculations described by each function using a specific value of x.
Define function composition and demonstrate composing functions with several examples. Explain that function composition is an operation that can be performed on two functions to form a third function (much like the operations of addition, subtraction, multiplication, and division). Emphasize the difference between f(g(x)) and g(f(x)) and explain that function composition is not commutative. Ask the student to compose functions and interpret the meaning of each composite function in context. Model using function notation appropriately.
Provide the student with verbal descriptions of realworld examples of functions and have the student write functions to describe the relationship among the variables. Ask the student to compose and interpret the functions in context. Give feedback as necessary.
If necessary, review function notation. Consider implementing the MFAS task What Is the Function Notation? (FIF.1.2). 
Moving Forward 
Misconception/Error The student is unable to compose functions correctly. 
Examples of Student Work at this Level The student writes functions that correctly model the given relationships but is unable to compose functions. For example, the student:
 Interprets composition as multiplication and writes f(g(x)) = 50(0.065x) and g(f(x)) = 0.065(x + 50).
 Writes that f and g are multiplied.

Questions Eliciting Thinking What does f(g(x)) mean? How is f(g(x)) different from g(f(x))?
Do you know what it means to compose functions? 
Instructional Implications Define function composition and demonstrate composing functions with several examples. Explain that function composition is an operation that can be performed on two functions to form a third function (much like the operations of addition, subtraction, multiplication, and division). Emphasize the difference between f(g(x)) and g(f(x)) and explain that function composition is not commutative. Ask the student to compose functions and interpret the meaning of each composite function in context.Â Provide additional opportunities to compose functions and interpret the composition in context.Â
Model using function notation appropriately.Â Correct any misuses of notation, if needed. Explain that the symbols chosen to name the function and to represent the independent variable x as the cost of the chair need to be used consistently throughout the problem. Provide explicit feedback to the student concerning any errors in notation that were made and guide the student to correct them. Provide continued opportunities to use function notation.Â
Consider implementing the MFAS taskÂ What Is the Function Notation?Â (FIF.1.2). 
Almost There 
Misconception/Error The student is unable to interpret the meaning of the composed functions. 
Examples of Student Work at this Level The student correctly writes and composes the functions but is unable to interpret the compositions in context. For example, the student:
 Does not even attempt to interpret the compositions.
 Provides an incorrect interpretation such as â€śitâ€™s a combination of the purchase price and the costâ€ť for f(g(x)) or g(f(x)).

Questions Eliciting Thinking What do you think 1.065x + 50 is calculating?
What do you think 1.065x + 53.25 is calculating? 
Instructional Implications Focus the studentâ€™s attention on the expressions that represent the compositions (e.g., 1.065x + 50 and 1.065x + 53.25). Review the meaning of x and the significance of 0.065 and 50 in the context of the problem. Guide the student to interpret each expression and what it calculates.
Correct any misuses of notation, if needed. Explain that the symbols chosen to name the function and to represent the independent variable x as the cost of the chair need to be used consistently throughout the problem. Provide explicit feedback to the student concerning any errors in notation that were made and guide the student to correct them. Provide continued opportunities to use function notation.
Consider implementing the MFAS task What Is the Function Notation? (FIF.1.2). 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student correctly:
 Writes f as f(x) = x + 50
 Writes g as g(x) = 1.065x or g(x) = x + 0.065x.
 Writes f(g(x)) =Â f(1.065x) = 1.065x + 50 or f(g(x)) = f(x + 0.065x) = x + 0.065x + 50 = 1.065x + 50 and explains the function as representing the cost of the chair with taxes applied plus the delivery fee.
 Writes g(f(x)) =Â g(x + 50) = 1.065(x + 50) = 1.065x + 53.25 or g(f(x)) =Â g(x + 50) = x + 50 + 0.065(x + 50) = x + 50 + 0.065x + 3.25 = 1.065x + 53.25 and explains the function as representing the cost of the chair and the delivery fee with taxes applied to both.
 Indicates that the function f(g(x)) represents a smaller cost because only the chair is taxed, not the delivery fee.

Questions Eliciting Thinking What do you think the graph of f(x) looks like? What do you think the graph of g(x) looks like? How do you know?
Is function composition commutative?
What is the relationship between the graphs of f(g(x)) and g(f(x))? 
Instructional Implications Provide the student with opportunities to write equations that define more complex functions and to write equations to define functions when given their graphs. 