Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 9 chapters and video availability01 · Read and understand
What you will learn
- Evaluate a composition and distinguish it from reversing the order.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Signed arithmetic, fraction and decimal operations, one-step equations, and introductory coordinate graphs.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Let f(x)=2x+1 and g(x)=x². Find f(g(3)) and g(f(3)).
Why this math matters
Model sequential conversions while tracking which transformation occurs first. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use the real-number system and the restrictions or data model stated in the question unless another domain is specified.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Follow the order of two function machines
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Let f(x)=2x+1 and g(x)=x². Find f(g(3)) and g(f(3)).
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
g(3)=9; f(9)=19
For f(g(3)), the inside rule acts first.
Develop the calculation
f(3)=7; g(7)=49
The reversed composition starts with a different intermediate value.
Check and interpret
f(g(3))=19; g(f(3))=49
Composition is generally sensitive to order.
The result
f(g(3))=19; g(f(3))=49
Composition is generally sensitive to order.
Common mistakes to catch
- Work from the inside outward.
- Preserve parentheses when an entire expression becomes a new input.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Write f(g(x)) for these rules.
Show a hint
Substitute x² into the input of f.
Reveal answer and explanation
2x²+1
The outer rule doubles the inner output and adds one.
Practice 2
Is g(f(x)) equal to 2x²+1?
Show a hint
Square the complete inner expression.
Reveal answer and explanation
No; it is (2x+1)²
The square applies to both parts of f(x), not only to the variable.
Take the idea with you
Model sequential conversions while tracking which transformation occurs first.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Compute a signed rate from two points
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