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Grade 9 · Functions

Calibration table: Preserve function-machine order

Calibration table investigation: preserve function-machine order. Change the quantities, follow the motion, and explain the result using the displayed mathematical relationship.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Calibration table: Preserve function-machine order. Stage: Input. Current value: -1. f(g(x)): 1. g(f(x)): 1Function order changes the calculationInput-1Square first1Then linear rule1Current stage value: -1Reversed order gives 1
Both functions are defined for all real inputs; no denominator or square-root domain restriction is hidden. Stage boxes represent ordered operations rather than a spatial graph. The readouts compare the same input in both orders.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Stage
Input
Current value
-1
f(g(x))
1
g(f(x))
1

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

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. This investigation begins with outer multiplier a = 1; outer shift b = 0; input x = -1. Treat the named setting as a constructed classroom model, then explain which relationships would still hold if its numbers changed.

A relationship to keep

f(u)=au+b, g(u)=u²; f(g(x))=ax²+b

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Represent the given quantities

    Begin with the selected input x and identify g as the inside function.

  2. STEP 2

    Follow the changing model

    Playback sends x through the square machine first, then applies the linear rule to that complete result.

  3. STEP 3

    Check and explain the relationship

    Reverse the machines and compare (ax+b)² with ax²+b. They can coincide at special inputs without being identical rules.

Your turn to explain

Make a prediction. Test your reasoning.

Calibration table: If f(u)=1u+(0) and g(u)=u², compare f(g(-1)) and g(f(-1)).

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

f(g(-1))=1; g(f(-1))=1. Work from the inner rule outward.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 9 · Algebra

Follow the order of two function machines

Evaluate a composition and distinguish it from reversing the order.