Grade 9 · Preserve function-machine order · 191 of 750
Preserve function-machine order · practice 20
Let f(u) = 4u + (-4) and g(u) = u². Find f(g(-3)) and g(f(-3)). Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Let f(u) = 4u + (-4) and g(u) = u². Find f(g(-3)) and g(f(-3)).
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
Evaluate the inner function first; changing order changes the input of the outer rule.
Work through the reasoning
Step 1
Translate the givens
For f(g(-3)), first g(-3) = (-3)² = 9.
Step 2
Calculate with the model
Then f(9) = 4 × 9 + (-4) = 32.
Step 3
Check and interpret
In reverse order, f(-3) = -16, then square: g(f(-3)) = (-16)² = 256. The outputs differ, demonstrating that composition order matters.
The answer
f(g(-3)) = 32; g(f(-3)) = 256.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
Composition feeds one function's output into the next function's input. The order matters because squaring after a shift is not the same operation as shifting after a square. Tracking the intermediate value prevents treating composition as multiplication of the rules. Starting quantities: Outer multiplier a = 4; Outer shift b = -4; Input x = -3.
f(u)=au+b, g(u)=u²; f(g(x))=ax²+b
03 · Reflect and transfer
Explain what changes and why.
Can two different composed rules agree at one input without agreeing everywhere?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.