Track the inner and outer steps, compare their order, and check whether the intermediate output is allowed.
Order the function machines
Paper starting model
Drag a handle or select an object, then tap its destination.
Input 2; order: f(u) = 2u → g(u) = u + 3.
Give your reason before checking.
Numeric readings
Values in this design
Machine order
x
Intermediate
Final output
f(u) = 2u → g(u) = u + 3
2
Send the input token
Send the input token
g(u) = u + 3 → f(u) = 2u
2
Send the input token
Send the input token
Tap and keyboard tools
Tab to an object. Arrow keys move handles; Enter selects a card or places it at a destination.
Select an object
Select a destination
Our saved board works
Works stay in this open page. Use the common notebook for consent-based saved notes.
Chapter 3 · Functions
Composition as two successive steps
Class ______ Name __________________ Date __________
Does swapping two function machines keep the answer the same?
My prediction and reason
Start with 2. Predict the result of adding 3 then doubling, and of doubling then adding 3, before running either machine.
Evidence and observations
For f(u) = 2u and g(u) = u + 3, record input, intermediate value and final output for both orders at input 2.
Repeat at inputs 0 and −3. Write the general formulas f(g(x)) and g(f(x)) and compare their difference.
Choose the projection’s “subtract 1, then square root” machines. Use g(x) = x − 1 followed by f(u) = √u; test x = 0, 1, 2 and record which intermediate value fails the outer machine.
Explain or give a counterexample
Transfer
Let g(x) = x + 1 and f(u) = 1/u. Find f(g(x)) and its domain. Then compare with g(f(x)); identify both the formula and domain differences.
Exit task
Let f(u) = √u and g(x) = x + 3. Find f(g(−2)) and the domain of f(g(x)).
Chapter 3 · Functions · Teacher reference
Composition as two successive steps
補充(第7課時) · 07_複合函數是依次完成兩個步驟.docx
5 min predict order effects; 12 min run and swap machines; 8 min derive formulas and domains; 12 min paper practice; 3 min exit check.
Does swapping two function machines keep the answer the same?
Start with 2. Predict the result of adding 3 then doubling, and of doubling then adding 3, before running either machine.
Evidence tasks
For f(u) = 2u and g(u) = u + 3, record input, intermediate value and final output for both orders at input 2.
Repeat at inputs 0 and −3. Write the general formulas f(g(x)) and g(f(x)) and compare their difference.
Choose the projection’s “subtract 1, then square root” machines. Use g(x) = x − 1 followed by f(u) = √u; test x = 0, 1, 2 and record which intermediate value fails the outer machine.
Reference reasoning
In f(g(x)), first apply g to x, then feed the whole output into f. The linear machines give f(g(x)) = 2x + 6 and g(f(x)) = 2x + 3, so order matters here. A composite input must lie in the inner domain and produce an intermediate value in the outer domain. The projection’s square-root task uses subtraction of 1: √(x − 1) requires x ≥ 1. The paper worked example uses a different constant, subtraction of 2: √(x − 2) requires x ≥ 2. The domain rule is the same, but the boundary changes with the inner rule.
Calculate f(g(2)) by recording the intermediate value.
f(u)=2u,g(u)=u+3;f(g(2))
First g(2) = 2 + 3 = 5.
Then f(5) = 2 × 5 = 10, so f(g(2)) = 10.
Return to the paper example: the inner rule subtracts 2. Find the composite rule and real domain.
f(u)=u,g(x)=x−2;f(g(x))
The whole inner output is x − 2, so f(g(x)) = √(x − 2).
Require x − 2 ≥ 0. The domain is [2, ∞).
Transfer and exit task
Let g(x) = x + 1 and f(u) = 1/u. Find f(g(x)) and its domain. Then compare with g(f(x)); identify both the formula and domain differences.
Let f(u) = √u and g(x) = x + 3. Find f(g(−2)) and the domain of f(g(x)).
g(−2) = 1, so f(g(−2)) = 1. The composite is √(x + 3), with domain [−3, ∞).
Computes the inner output before the outer square root.
Restricts the intermediate value x + 3 rather than excluding all negative starting inputs.
Questions
Which machine receives the original input in f(g(x))?
Why is the square-root boundary 1 in the projection task but 2 in the paper example?
Expected responses
Students may multiply f and g. Trace the arrow through two distinct stages.
Students may apply the outer restriction directly to x. Keep the intermediate box visible.
Confusions
Do not reverse the order written inside f(g(x)).
One input with different outputs refutes equality; a general algebraic comparison explains the rule for all inputs here.
Board plan
x → g(x) → f(g(x)).
The whole inner output becomes one input to the outer rule.
Composite domain: inner input allowed and intermediate output allowed.
