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Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Find the allowed inputs of a composition before simplifying it.
- Justify the conclusion "The domain is (2,∞)" using the stated assumptions.
Before you start
Square roots, reciprocal functions, and domains.
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
For f(x)=√x and g(x)=1/(x−2), find the domain of f(g(x)).
Why this math matters
Find the allowed inputs of a composition before simplifying it. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Functions are real-valued.
- Square roots mean nonnegative roots.
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.
Track a domain through two function machines
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For f(x)=√x and g(x)=1/(x−2), find the domain of f(g(x)).
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.
Build the model
f(g(x))=√(1/(x−2))
The output of g must be an allowed input of f.
Work through the mathematics
Require x≠2 and 1/(x−2)≥0
The denominator cannot vanish and the square-root input must be nonnegative.
Check the conclusion
The domain is (2,∞)
A positive numerator makes the reciprocal positive exactly when x−2 is positive.
The result
The domain is (2,∞)
A positive numerator makes the reciprocal positive exactly when x−2 is positive.
Common mistakes to catch
- Function composition generally depends on order.
- Simplification must not erase an excluded 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
Find the domain of g(f(x)).
Show a hint
Require x≥0 and √x≠2.
Reveal answer and explanation
[0,∞) excluding 4
Reversing the composition changes its restrictions.
Practice 2
Is f(g(3)) defined?
Show a hint
Evaluate the inside function first.
Reveal answer and explanation
Yes; its value is one
g(3)=1 and √1=1.
Take the idea with you
Describe a two-stage measurement conversion whose first output must fit the second stage's input range.
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: Choose a branch before inverting a quadratic
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