What quantity did you change?
Labs / Mathematics
Composition Machine
Pick two functions and watch a value ride the pipeline: it passes through g, comes out changed, then flows into f. The graph plots the whole composite f(g(x)), and you can scrub the input by dragging across it. Composition means chaining functions together — the output of one becomes the input of the next.
Labs / Mathematics / Composition Machine
Composition Machine
Pick two functions and watch a value ride the pipeline: it passes through g, comes out changed, then flows into f. The graph plots the whole composite f(g(x)), and you can scrub the input by dragging across it. Composition means chaining functions together — the output of one becomes the input of the next.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
inner g(1.20) = 1.44result f(g(x)) = 2.88y = 2·(x²)
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- Set g to x² and f to 2x. Does f ∘ g give the same curve as g ∘ f, or does order matter?
- Drag the input across zero. Which functions flip the sign of the result, and which never do?
- Turn on Animate sweep. Where is the composite curve steepest, and what does that tell you about how fast the output changes?
- Choose g = sin x and f = x². Why does the composite stay trapped between 0 and 1?
- Make both machines −x. Two sign flips — where does the value end up, and what does that say about doing something twice?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.